On the local Frattini subgroup of a finite group

Let $G$ be a finite group and let $π(G)$ be the set of prime divisors of $|G|$. For each $p\in π(G)$, choose a Sylow $p$-subgroup $G_p$ of $G$. The subgroup $Ψ(G)= \prod_{p\in π(G)} Φ(G_p)_G$ is called the \textit{local Frattini subgroup} of $G$. Here, for $J \leqslant G$, the symbol $J_G$ denotes t