Automizers as extended reflection groups

Let G be a finite group having an abelian Sylow p-subgroup P. Broué, Malle and Michel have shown that if G is a simple Chevalley group, then the automizer of P is an irreducible complex reflection group (for p not too small and different from the defining characteristic).The aim in this note is to show that a suitable version of this property holds for general finite groups.