On Jan. 23, 1933, Congress passed the 20th Amendment to the U.S. Constitution, which moved the president-elect’s Inauguration Day from March 4 to January 20.
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My guess is for the date of inauguration to be changed only congress could change it.
20th Amendment Section 3 stipulates what happens if the president-elect fails to qualify before Inauguration Day:
“If, at the time fixed for the beginning of the term of the President, the President elect shall have died, the Vice President elect shall become President. If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified; and the Congress may by law provide for the case wherein neither a President elect nor a Vice President elect shall have qualified, declaring who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified.“