It isn't up to the president to make sure the next one was legitimately elected. He has no power to stop the transfer once the Electoral College votes are certified by Congress. it's their job to make sure the process was legit.
What happens if they don’t do their job/ are just as compromised?
We're witnessing that right now. Despite plenty of evidence (much of which that was dismissed out of hand) a contested election was certified and the "winner" will be the next to hold office.
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