Equivariant Alperin-Robinson's 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 Alperin's 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 an Appendix we develop some old arguments that we need in the proof

On the reduction of Alperin's Conjecture to the quasi-simple groups

We show that the refinement of Alperin's Conjecture proposed in "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, can be proved by checking that this refinement holds on any central k*-extension of a finite group H containing a normal simple group S with trivial centralizer in H and p'-cyclic quotient H/S. This paper improves our result in [ibidem, Theorem 16.45] and repairs some bad arguments there