Gauntlet is mostly between master and apprentice - although many houses have specific traditions about gauntlet.
The master decides when the apprentice is ready for the gauntlet, sets the gauntlet, and determines if the apprentice passed or not. Usually the gauntlet is after 15 years, but it doesn't have to be.
It depends on the house, and for few houses is it spelled out.
Flambeau is one of the few for which it is, and interestingly, the same gauntlet applies whether it is graduating apprentice or a full-fledged magus who wants to join the house. They like to make the gauntlets as public as possible, and it takes a sponsor who is a member of the house to set the gauntlet. It is reasonable to expect that as far as Flambeau are concerned, a new gauntlet with a valid sponsor would be set, and at that point it matters little if it is an apprentice, a hedge wizard, or an existing member of the Order.
YHMV