Autohover yes. It has a GTC (H) function that allows the pilot to either set a ground vector or ‘double tap’ for a hover. As for auto transition, no it does not, at least the RAF Jupiter does not - maybe it is an option?
So it would seem that the Jupiter is hardly more capable than the Wessex or the 412 that preceded it, SAR modes on the AFCS were what was lacking on both aircraft and for £145 million I'd expect an aircraft that has SAR as part of its role there to have that fitted.