SevenOP,
As our Public Service Broadcaster, the BBC is obliged to show a quota of Current Affairs programmes, and QuestionTime is fulfilling that remit.
It was originally a BBC in-house production but the Thatcher Conservative Government decreed that the BBC must buy more television made by independent producers, so Question Time was hived off.
A former BBC employee Brian Leppings set up a production company, and they were awarded the contract to make the programme.
So the BBC is obliged to make QuestionTime, but has to commission an independent company to make it, and pay the company from the licence fee. It is a BBC programme made by outside contractors. Most of BBC programmes are made by third party contractees.