On finite groups admitting automorphisms with nilpotent centralizers

Let p be a prime. Let A be a finite group and M be a normal subgroup of A such that all elements in A∖M have order p. Suppose that A acts on a finite p′-group G in such a way that CG(M)=1. We show that if CG(x) is nilpotent for any x∈A∖M, then G is nilpotent. It is also proved that if A is a p-group and CG(x) is nilpotent of class at most c for any x∈A∖M, then the nilpotency class of G is bounded solely in terms of c and |A|.