According to FCL.045 a pilot with a valid Licence must always carry it with them (plus medical, plus photo ID) when exercising that Licence's privilege(s) (ie flying an aircraft).
Someone who does not have a valid Licence must be a student and they can only fly solo if authorised, and supervised, by an FI (using their privilege FCL.905.FI (a))
However I would be unsure of the wisdom of authorising someone to fly solo back from their PPL Skill Test (generally driving instructors don't let candidates drive back from a driving test) - too many emotions possible.
Even a Licenced pilot will probably be a student again at some point (revision, rating renewal, new rating, etc) but, as they hold a valid Licence, then an FI's privilege FCL.095.FI (a) no longer applies. There has been a long discussion previously on this forum on whether/or not a valid Licence holder with an expired Rating can be authorised to go solo and, if so, by whom. (I hold that it is not possible).