What's happening in Parliament this week?

It's the last week of term, when, by ancient Westminster tradition, MPs are allowed to bring in board games.

They may need something to amuse them, because much of the week's business will see each House waiting on the other, as they seek to finalise the details of a series of measures, notably the highly contentious Illegal Migration Bill.

The expectation is that the outstanding issues on this bill (where, remember, the government was defeated 20 times in the Lords) will all be resolved before the break.

But if not, the process of parliamentary "ping pong" - when bills bounce between the Commons and Lords until the final wording is agreed - will resume in September.

So far, it looks like Illegal Migration, Social Housing (Regulation), and Strikes (Minimum Service Levels) will get through next week, while Northern Ireland Troubles (Legacy and Reconciliation) will not be finished until September.

The Illegal Migration Bill has produced some remarkable moments.

Former prime minister Theresa May defied a three-line whip to vote against the government to preserve the modern slavery protections her government brought in.

It's also been calculated that the 18 Commons divisions held on Tuesday night, requiring honourable members to plod through 18 circuits of the voting lobby, added 7,500 paces to the daily step count of the average MP.

Meanwhile, another and perhaps even bigger ping-pong extravaganza looms on the Levelling Up Bill, where report stage in the Lords has already produced government defeats in two days of debate, on subjects like levelling-up funding, child poverty, health disparities, and rural issues.

This is what happens when a bill is so wide-ranging that almost any domestic policy issue can be crowbarred into it. Opposition sources speculate there could be as many as 30 government defeats by the time the bill bounces back to MPs in the autumn.

The end of term is often a moment for internal matters, and on Wednesday MPs will debate a significant tightening of the rules for all-party parliamentary groups.

There are hundreds of these cross-party organisations of MPs and peers, on topics ranging from diseases to sports, to cycling and beer.

Some are little more than social organisations, others have serious influence on policy and legislation - a good recent example is the APPG on Autism, under the late Dame Cheryl Gillan, which saw a private member's bill passed and then pushed ministers for action to help people with autism.

The concern is that some APPGs may act as a conduit for outside interests, including foreign governments, to influence Parliament or the UK government.

So new rules are being proposed to sharpen MPs' responsibility for the running of APPGs, with the focus on those groups receiving financial benefits.

Those APPGs will be required to register them, and the new rules would ban foreign governments from financing the secretariats of APPGs (MPs who serve as officers of an APPG will be expected to exercise due diligence in checking where their funding comes from).

In the Lords, some of those who received peerages in Boris Johnson's resignation honours list will take their seats, starting with his former political secretary Ben Gascoigne and former London mayoral candidate Shaun Bailey when the Lords sits on Tuesday.

On Thursday, it will be the turn of former Conservative Party political director Ross Kempsell, and Kulveer Ranger, who was Mr Johnson's director of transport policy when he was mayor of London.

Finally, with the end of term in sight, committee action is concentrated on Monday and Tuesday, and there are some important select committee reports to watch out for.

On Tuesday, the Public Accounts Committee reports on the awarding of Covid contracts to PPE Medpro, following a Commons vote last year calling on the government to release correspondence, documents and advice to the committee.

And then there's the publication of the Defence Select Committee's six-month inquiry into the UK's defence procurement system, which was led by Conservative former minister and sub-committee chair Mark Francois.

Having watched some of the hearings, it's hard to imagine they're going to conclude that all's well, and Mr Francois is an experienced parliamentary guerrilla, who can be expected to follow up their findings with continued pressure on ministers - watch out for an intervention at PMQs if Mr Speaker gives him the chance.

Elsewhere, the Transport Committee has re-opened its call for evidence for its inquiry into accessible transport, after train operating companies announced plans to close over 300 ticket offices across the rail network.

Commons: (14:30) Education Questions, with any urgent questions or government statements following at 15:30.

Main debate: MPs respond to Lords amendments to the Illegal Migration Bill - where peers are still pressing nine issues, including child detention, protection for LGBT refugees, safe and legal routes for asylum seekers, and more. There will also be ping-pong on the Strikes (Minimum Service Levels) Bill - where the main outstanding issue is on the consultation required before minimum service levels are set. There will be an hour set aside for each bill.

Westminster Hall: (16:30) MPs consider a petition urging the government to give teaching assistants a pay rise.

Committees: Levelling Up, Housing and Communities (16.00) ends its inquiry into financial reporting and audit in local authorities, with questions to local government minister Lee Rowley. With more councils in effect going bankrupt, they will ask whether there is an auditing crisis in England's councils and examine the future role of the Auditing, Reporting and Governance Authority (ARGA).

Lords: (14:30) It's the last of four days of report stage debate on the Online Safety Bill. No votes are expected. There's also a slot to allow peers to consider the latest Commons amendments to the Illegal Migration Bill.

Commons: It's Foreign Office Questions at 11:30. The DUP's Jim Shannon will present a ten-minute rule bill to require a report on increasing the income threshold for the high-income Child Benefit Charge in line with inflation. Labour's Richard Burgon will present a bill to ban MPs from receiving financial or other benefits from oil and gas companies. This will happen without debate, but there's an interesting sense of a marker going down.

Main debate: Further consideration of more Lords amendments to assorted bills, including the Northern Ireland Troubles (Legacy and Reconciliation) Bill, where the government lost votes on immunity from prosecution for those who co-operate with the proposed new mechanism for investigating historic crimes, and on minimum investigation standards. Peers are not scheduled to respond to any changes made by the Commons until September.

Committees: Public Administration and Constitutional Affairs (10.00) looks at the government's new smartphone-based public warning system with its siren-like emergency alerts, after the nationwide test in April failed to reach an estimated 7 per cent of compatible devices.

The Lords Communications Committee (14:30) questions BBC Director-General Tim Davie and acting chair Dame Elan Closs Stephens on BBC leadership. As they delicately put it: "In light of recent events, what concerns have been raised about the adequacy of the BBC's governance arrangements, and how it is addressing these?"

Lords: (14:30) It's day three of seven of report stage debate on the Levelling-Up and Regeneration Bill, with votes expected on the devolution of bus services, an attempt to scrap the proposed new Infrastructure Levy (over concerns that it will not do enough to provide affordable housing), and a new clause on the governance of national parks from Conservative peer Lord Randall.

Peers are also preparing for a long night processing Commons amendments to the Illegal Migration Bill, with restaurants and bars opening late, and hot snacks available until half an hour after the close of business.

Commons: (11:30) Science, Innovation and Technology Questions, followed at noon by Prime Minister's Question Time.

Main debate: More ping-pong with the Lords, followed by a debate on proposed new rules for all-party parliamentary groups (see above).

Committees: Health (10.00) continues its future cancer inquiry with evidence on the new cutting-edge technologies that could boost earlier cancer diagnosis, including the NHS-Galleri blood test, which could change cancer screening. Home Affairs (10.00) concludes its inquiry into human trafficking with evidence from safeguarding minister Sara Dines on how potential victims of trafficking and modern slavery are supported.

Lords: (15:00) It's day five of five of report stage debate on the Online Safety Bill. If the schedule holds and peers complete report consideration today, the bill will have its third reading immediately after they return from their summer recess in early September. There's also another slot available for consideration of Commons amendments to the Illegal Migration Bill.

Commons: (09:30) Culture, Media and Sport Questions, followed by mini-question times for the MPs who speak for the Church Commissioners, the House of Commons Commission (the administrative body of the House), the Public Accounts Commission, the Restoration and Renewal Client Board (which represents MPs and peers' views in the mega-project to revamp Parliament's main buildings) and the Electoral Commission.

Then comes the weekly update on the forthcoming Commons agenda, from the Leader of the House - this time setting out what MPs will be debating when they return after recess.

Main debate: It's the Sir David Amess summer adjournment debate - the traditional end of term debate, in which any MP can raise any subject they choose, named in honour of the murdered veteran MP.

Lords: (11:00) Day four of seven of report stage on the Levelling-Up and Regeneration Bill. Watch out for amendments on the continuation of high street financial services - banks, post offices and cash machines.

Neither House sits on Friday.