  We work towards a version of generalized Harish-Chandra theory compatible
with Clifford theory and with the action of automorphisms on irreducible
characters. This provides a fundamental tool to verify the inductive conditions
for the so-called local-global conjectures in representation theory of finite
groups in the crucial case of groups of Lie type in non-defining
characteristic. In particular, as shown by the author in an earier paper, this
as a strong impact on the verification of the inductive condition for Dade's
Conjecture. As a by-product, we also show how to extend the parametrization of
generalized Harish-Chandra series given by Brou\'e, Malle and Michel to the
non-unipotent case by assuming maximal extendibility.
