What might Trump be doing during Biden's inaugural address?

First of all, he gets a pass if there's no in-person address. And even if there is, it will be sparsely attended because of Covid, so we know Trump will make disparaging remarks about Biden's attendance numbers forever.

Donald Trump is discussing different ways to disrupt the impending Joe Biden era, chief among them by announcing another run against him.

According to three people familiar with the conversations, the president, who refuses to acknowledge he lost the 2020 election as he clearly did, has not just talked to close advisers and confidants about a potential 2024 run to reclaim the White House but about the specifics of a campaign launch. The conversations have explored, among other things, how Trump could best time his announcement so as to keep the Republican Party behind him for the next four years. Two of these knowledgeable sources said the president has, in the past two weeks, even floated the idea of doing a 2024-related event during Biden’s inauguration week, possibly on Inauguration Day.

  Daily Beast

He'll do SOMEthing to call attention to himself that day.

The president and some of his closest associates have already started surveying prominent donors to get a sense of who would be with him, or perhaps against him, if he chose to run in the 2024 election. Some top Trump allies have told The Daily Beast that they are doing what they can to stay in the president’s good graces, calculating that doing so will help ensure a seat at the table and a future in the party—in the event he runs again.

Hitch your wagon to a batshit crazy loser. Sounds like a great plan.

According to two sources with direct knowledge of the matter, the president has privately bragged that he’d still remain in the spotlight, even if Biden is in the Oval Office, in part because the news media will keep regularly covering him since—as Trump has assessed—he gets the news outlets ratings and those same outlets find Biden “boring.”

[...]

On Thursday, Bloomberg reported that during an Oval Office meeting earlier this month between Trump, National Security Adviser Robert O’Brien, Vice President Mike Pence, and Secretary of State Mike Pompeo, the president said he planned on running in 2024, if the 2020 election results were not nullified by Trump’s attorneys.

“If you do that—and I think I speak for everybody in the room—we’re with you 100 percent,” O’Brien told the president, according to the Bloomberg report.

Oh, I bet Mike Pence isn't. He'll be wanting to run on his own.  And I wouldn't be surprised if Mike Pompeo wants to do the same.

Trump’s refusal to concede has been supplemented by an attempt to make it harder for Biden to reverse his policy achievements. Indeed, even before Election Day 2020, various Trump officials working in the administration were charting paths forward to make it harder for Biden to reenter the Iran nuclear agreement.

So, it's just a coincidence, I'm sure, that Pompeo recently secretly met with Mohammed bin Salman and Benjamin Netanyahu, and few days later a major Iranian nuclear scientist was assassinated. 

 ...but hey, do what you want...you will anyway.