$p$-central action on 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 $p=2$) in a specified subgroup of $G$, then both $A$ and $[G,A]$ behave regularly, that is the elements of order dividing any power $p^i$ in each one of them form a subgroup; moreover $A$ and $[G,A]$ 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 problems 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 $p$ ($4$ if $p=2$) in a given term of the lower central series of $P$ lies in the center of $P$.

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