Fake Galois Actions

We prove that for all non-abelian finite simple groups $S$, there exists a fake mth Galois action on IBr$(X)$ with respect to $X⊲X⋊$ Aut$(X)$, where $X$ is the universal covering group of $S$ and $m$ is any non-negative integer coprime to the order of $X$. This is one of the two inductive conditions needed to prove an $\mathrm{\ell }$-modular analogue of the Glauberman-Isaacs correspondence.