  Let $G$ be a finite group isomorphic to $SL_n(q)$ or $SU_n(q)$ for some prime
power $q$. In this paper, we give an explicit description of the action of
automorphisms of $G$ on the set of its irreducible complex characters. This is
done by showing that irreducible constituents of restrictions of irreducible
characters of $GL_n(q)$ (resp. $GU_n(q)$) to $SL_n(q)$ (resp. $SU_n(q)$) can be
distinguished by the rational classes of their unipotent support which are
equivariant under the action of automorphisms. Meanwhile, we give a criterion
to explicitly determine whether an irreducible character is a constituent of a
given generalized Gelfand-Graev character of $G$. As as application, we give a
short proof of the global side of Sp{\" a}th's criterion for the inductive
McKay condition for the irreducible characters of $G$.
