I think in spite of any executive order he would issue, Obama is required by law to leave office on the 20th of January at exactly noon - his term is up and he’s gone - there is a little known episode in US history - I don’t remember all the details, but it was back in the 1800’s when presidential transfer was about to take place and for some reason the incoming individual was not available immediately to take the oath - the man whose term was up left office and for about four hours someone else - maybe the secretary of state - was officially the president until the newly elected man arrived to be sworn in - time’s up at noon on inauguration day for Obama.....
Actually the guy’s name was David Atchison, senator from Missouri and president pro-tem of the Senate - inauguration day for Zach Taylor was March 4, 1849, a Sunday - but Taylor refused to be sworn in on a Sunday - previous president Polk left office officially at noon on the 4th, so for one day perhaps someone else was president - not clear if it was Atchison though since his term had also expired on the 4th - in any event he spent most of the time sleeping, so never knew he might have been president until later.....