$\DeclareMathOperator\Syl{Syl}$Let $G$ be a finite group, $\Phi(G)$ is the Frattini subgroup of $G$. Let $P\in \Syl_{p}(\Phi(G))$, where $p\in \pi(\Phi(G))$ and  $\pi(\Phi(G))$ is the set of prime divisors  of $|\Phi(G)|$. Suppose every subgroup $H$ of $P$ of order $p$ is normal in $G$. Let $P_1\in \Syl_{p}(G)$, we can use $|H|=p$ and $H$ is normal in $G$ to prove that $H\leq P_1$. Then can we prove $H\leq Z(P_1)$?