well, they gained control by seizing the canal then lost it due to the invasion, hence why I said 'regain'. Both the Soviet and US pressure resulted in the stop of the invasion, but certainly the USSR had an important role, if anything to distract the media from its invasion of Hungary.
Not egypt, but the Suez Canal company was was not the legitimate owner of the canal.
Interestingly, it was the disapproval of the US under Eisenhower that stopped the military action of Britain and France that ended their intervention. https://en.m.wikipedia.org/wiki/Suez_Canal_Company