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

Action of automorphisms on irreducible characters of groups of type \textsf{A}