Fake Galois actions

We prove that for all non-abelian finite simple groups [Formula: see text], there exists a fake [Formula: see text]th Galois action on [Formula: see text] with respect to [Formula: see text], where [Formula: see text] is the universal covering group of [Formula: see text] and [Formula: see text] is any non-negative integer coprime to the order of [Formula: see text]. This is one of the two inductive conditions needed to prove an [Formula: see text]-modular analogue of the Glauberman–Isaacs correspondence.