An analogue to Glauberman's $ZJ$-theorem

Let $P$ be a finite $p$-group, $p$ an odd prime. Using certain versions of $p$-stability it is shown that there exists a nontrivial characteristic subgroup $W$ in $P$ that is normal in every finite $p$-stable group $G$ satisfying ${C_G}({O_p}(G)) \leq {O_p}(G)$ and $P \in {\text{Sy}}{{\text{l}}_p}(G)$. Moreover, $W$ contains every abelian subgroup of $P$ normalized by $W$.