Rishi Sunak may yet find himself asking what Sir Gavin Williamson would have done between now and Christmas, as he faces an unenviable uphill struggle to manage his party. 

While the prime minister presented himself as a fresh start when he entered Number 10 two weeks ago, the fissures, fury and fire inside the Conservative party have only partially calmed. This may yet change given the scale of the challenge ahead.

Here are five of them…

The Autumn Statement

The storm clouds are gathering. A revolt is brewing on tax. The Conservative membership refused to put Mr Sunak in Downing Street over the summer – in part – because as chancellor he took Britain to a 70-year high. Mr Sunak has subsequently been installed without consulting them.

Everything has changed in the interim, with Trussonomics causing a spike in debt costs and a run on the pound, but at the same time, nothing has changed. And the Tory right is already agitating.

On Thursday evening, Iain Duncan Smith told Sky News that “it will be of deep concern if we go over the top on tax rises because it’s absolutely a fact of life that tax rises will make the recession deeper”.

More on Rishi Sunak

He thinks the size of the black hole, which he stresses is based on forecasts (which can be wrong), may be smaller than expected particularly if energy costs drop. Newspapers have speculated on tax rises on every front, from income to inheritance tax. When the latter was floated at the weekend, one Tory MP told me: “You do have to ask what is the point of a Tory government anymore.”

A quickie Brexit deal on Northern Ireland

This is a sleeper cell issue that could cause untold pain to Mr Sunak’s premiership if he gets it wrong. Some MPs believe there is a chance of a deal within weeks to overhaul the Northern Ireland Protocol – the agreement between the UK and EU preserves both the integrity of the EU single market as well as the lack of a hard border on the island of Ireland.

US President Joe Biden has suggested a deadline for a deal is the 25th anniversary of the Good Friday Agreement next April, and he is personally keen to resolve what he regards as a long-running sore. In his opening fortnight, Mr Sunak has made a huge effort to repair relations with the French and US presidents.

He was not an instinctive hardliner in EU Brexit discussions in Boris Johnson’s government. You can see the political attraction of a deal for Mr Sunak. But again, beware the hard Brexiteers. They – and some in the Unionist community in Northern Ireland – fear what will emerge is an expedient deal to reduce checks which still leaves this part of the UK subject to the jurisdiction of the European Court of Justice.

But if this happens, what happens to Suella Braverman, the home secretary, Chris Heaton Harris, the Northern Ireland secretary and Steve Baker, a Northern Ireland minister, and the chance of a revolt on his right flank?

Mr Duncan Smith told Sky News that the EU was not in a position to do a decent deal with the UK, and Mr Sunak should wait until after controversial legislation giving the UK the unilateral ability to disapply the protocol is in place. Theresa May, Boris Johnson and Liz Truss never successfully took on the ERG wing of the Tory party: will Mr Sunak be different?

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Analysis: ‘PM wasn’t as on form’

The trial of Boris Johnson

Forget “I’m A Celebrity”, the reality TV drama of late autumn could be the Privileges Committee inquiry into Mr Johnson over whether statements by the ex-PM during partygate “appear to amount to misleading the House”.

Although not formally confirmed, the plan has been to televise parts of the hearings – in effect to allow viewers to watch the evidence themselves in the hope that they reach the same conclusion as the panel of MPs chaired by Harriet Harman.

This has been plenty of potential to upend Mr Sunak’s administration. The PM’s spokesman said this week that this government “takes our responsibilities to assist the committee seriously” – a far cry from the approach taken when Ms Truss was in power.

Expect diaries, WhatsApp messages that have survived, and testimony all to be shared. Will Boris Johnson-supporting MPs be relaxed about the active decision by the Mr Sunak government to be complicit in the investigation?

Or will those who transferred their allegiance from Mr Johnson to Ms Truss – and now find themselves in the wilderness – allow themselves to become riled up at events?

Read more: Has Rishi Sunak’s baptism of fire showed opposition leaders make the best PMs?

By-elections and opinion polls

As this parliament enters the twilight years, Mr Sunak cannot avoid multiple potentially bruising electoral tests. The first couple were in seats that elected Labour MPs last time around: on 1 December the City of Chester goes to the polls in a by-election to find a successor to Christian Matheson, who resigned after sexual misconduct allegations which he denied.

There will also be a by-election in Stretford and Urmston to find a replacement for Kate Green, the Labour MP who is stepping down to become deputy mayor of Manchester. But this will be little sweat for Mr Sunak since Labour will be expected to win. But then the party could be facing the self-inflicted wound generated by peerages for two Mr Johnson allies in his resignation honours list could cause by-elections.

Nadine Dorries, the mid-Bedfordshire MP with a 24,664 majority and Nigel Adams, the Selby and Ainsty MP which has a 20,137 majority could both trigger much more problematic elections. The margin of Tory victory in 2019 in both was considerable: but that means losses in these two very different parts of the county will be cataclysmic.

Rishi Sunak, Paul Dacre and The Blob

This is niche – but high impact. Ex-Daily Mail editor Paul Dacre is tipped to feature on Mr Johnson’s resignation honours list for a peerage. This would come despite the recommendation of House of Lords appointments committee, which reportedly rejected his nomination earlier in the year.

However, it would be for Mr Sunak to ultimately push through Mr Johnson’s recommendation against what Mr Dacre himself might term “the blob”. Would he risk the likely criticism for handing a peerage in the face of the rejection by the official committee, or would he risk the wrath of the Daily Mail by refusing?