All special actions from all ships are cleared at the end of the end phase. They may go away earlier than that in some circumstances, for example:
"Activate jump engines" goes away immediately when the ship that made that special action goes through the jump point, and the jump point closes. You cannot make another SA of course, since SA's are (meant to be) declared at the start of that ship's movement.
All special actions will end immediately if a ship becomes skeleton crewed (and has no flight computer), or receives a "no special actions" crit.
You might as well remove SA's for All Power To Engines and Come About immediately (or just not bother putting tokens down) since they don't affect anything after movement phase anyway.