Action of automorphisms on irreducible characters of symplectic groups

Assume G is a finite symplectic group over a finite field of odd characteristic. We describe the action of the automorphism group on the set of ordinary irreducible characters of G. This description relies on the equivariance of Deligne–Lusztig induction with respect to automorphisms. We state a version of this equivariance which gives a precise way to compute the automorphism on the corresponding Levi subgroup; this may be of independent interest. As an application we prove that the global condition in Späth's criterion for the inductive McKay condition holds for the irreducible characters of .