Extensions of unipotent characters and the inductive McKay condition

We prove that unipotent characters of groups of Lie type extend to their inertia groups in the full automorphism group. We then apply this in order to show that certain groups of Lie type are good for the prime p=3 or p=2. This deals with all the exceptions not covered by our general approach for verifying the McKay conjecture for all groups of Lie type in non-defining characteristic.