Abstract Let đș be a finite reductive group such that the derived subgroup of the underlying algebraic group is a product of quasi-simple groups of type đ . In this paper, we give an explicit description of the action of automorphisms of đș on the set of its irreducible complex characters. This generalizes a recent result of M. Cabanes and B. SpĂ€th [Equivariant character correspondences and inductive McKay condition for type đ , J. Reine Angew. Math. 728 (2017), 153â194] and provides a useful tool for investigating the local sides of the local-global conjectures as one usually needs to deal with Levi subgroups. As an application we obtain a generalization of the stabilizer condition in the so-called inductive McKay condition [B. SpĂ€th, Inductive McKay condition in defining characteristic, Bull. Lond. Math. Soc. 44 (2012), 3, 426â438; Theorem 2.12] for irreducible characters of đș. Moreover, a criterion is given to explicitly determine whether an irreducible character is a constituent of a given generalized GelfandâGraev character of đș.