Normally acting up would be agreed for a defined period of time, additional allowance (if any) would be specified and what happens at the end of the period would also be stated, usually a return to the permanent position.
Depends how long 'so long' is, and why it was acting up in the first place. Acting up would normally be to fill a gap while recruiting or similar. If no end is in sight and no attempt being made to do anything else person would have an argument. Not a legal one, just a good one.
Most organisations would/should then advertise the position as permanent, and acting up person would be appointed on permanent basis.