p-Central action on finite groups

Let G be a finite p-group acted on faithfully by a group A. We prove that if A fixes every element of order dividing p   (4 if p=2p=2) in a specified subgroup of G, then both A   and [G,A][G,A] behave regularly, that is the elements of order dividing any power pipi in each one of them form a subgroup; moreover A   and [G,A][G,A] have the same exponent, and they are nilpotent of class bounded in terms of p and the exponent of A. This leads in particular to a solution of a problem posed by Y. Berkovich. In another direction we discuss some aspects of the influence of a p-group P on the structure of a finite group which contains P as a Sylow subgroup, under assumptions like every element of order dividing p   (4 if p=2p=2) in a given term of the lower central series of P lies in the center of P.