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In reply to the discussion: Trump says he'll still run if convicted and sentenced, responds to security footage allegations [View all]ExWhoDoesntCare
(4,741 posts)Nothing says it has to happen in a particular place. Only that it has to happen at noon on 20 January (4 March before FDR), or soon after a President has died or had to step down.
But it can happen anywhere.
LBJ got sworn in on Air Force One, in perhaps the most famous inauguration photo, ever:

Two incidents of inclement weather forced the swearing in to another location. Taft had his inauguration in the Senate chamber in 1909. Blizzard conditions forced reagan to have his swearing in at the Capital Rotunda in 1985.
Three inaugurations happened in the White House--all modern era: FDR & Truman in 1945, for obvios wartime safety reasons, and then Gerald Ford in 1974.
So if a POTUS-elect had to take the oath of office in prison, it would be arranged.