Equivariance and extendibility in finite reductive groups with connected center

We show that several character correspondences for finite reductive groups are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to has connected center. The correspondences we consider are the so-called Jordan decomposition of characters introduced by Lusztig and the generalized Harish-Chandra theory of unipotent characters due to Brou,-Malle-Michel. In addition we consider a correspondence giving character extensions, due to the second author, in order to verify the inductive McKay condition from Isaacs-Malle-Navarro for the non-abelian finite simple groups of Lie types , and .