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

We show that the refinement of Alperin's Conjecture proposed in L. Puig (2009) [10, Chap. 16], 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 L. Puig (2009) [10, Theorem 16.45] and repairs some bad arguments there.