Equivariant Alperin–Robinsonʼs Conjecture reduces to almost-simple k⁎-groups

In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions. Our purpose here is to show that the equivariant refinement of the Alperinʼs Conjecture for blocks formulated by Geoffrey Robinson in the eighties can be reduced to checking the same statement on any central k⁎-extension of any finite almost-simple group, or of any finite simple group up to verifying an “almost necessary” condition. In Appendix A we develop some old arguments that we need in the proof.