Groups of automorphisms of finite $p$-groups

Thompson [1] showed that if $p$ is an odd prime number, $A$ is a $p$-group of operators of the finite group $P$ in which the Frattini subgroup $\Phi(P)$ is elementary and central, and $P/\Phi(P)$ is a free $Z_pA$-module, then $C_P(A)$ covers $C_{P/\Phi(P)}(A)$. Then he proposed the question of whether it is possible in this theorem to weaken the hypothesis that $\Phi(P)$ be elementary and central. In the work it is shown that this hypothesis may be replaced by a much weaker one; it is sufficient that P be met-Abelian and have nilpotence class prime-subgroups of Sylowizers of a $p$-subgroup of a solvable group [2].