A Newsweek article: “How Trump Could Lose the Election– and Still Remain President.” Is it possible for Trump, if he loses the election, to claim election fraud and have the results thrown out so that the red state legislatures to vote Trump in with the 12th amendment?(https://www.newsweek.com/how-trump-could-lose-election-still-remain-president-opinion-1513975) The are articles that say otherwise. (https://takecareblog.com/blog/red-state-legislatures-cannot-cancel-the-upcoming-presidential-election) Is it possible?
We are talking about a scenario where the outcome is under dispute. As the incumbent President would see it, he would not have been voted out of office, there would merely be 'fake news' peddling that theory.
There are always little irregularities when millions of voters go to the polls. In a stable, functioning democracy, there is a generally accepted, legal process to sort these out, and there is an unspoken assumption that "honest mistakes" will average out. It becomes problematic if the irregularities are favoring one side over the other or even if it just looks that way.
Trump cannot stay in office if a clear majority oof the political system in the US concludes that he has lost. Courts would rule against him and those rulings would be enforced by the government agencies.
He might stay in office if the election outcome is unclear by January 2021. If some courts support him, some oppose, and many are deadlocked, there could be no momentum for enforcing a handover.