On the local Frattini subgroup of a finite group

Let $G$ be a finite group and let $\pi (G)$ be the set of prime divisors of $\mathrm{\mid }G\mathrm{\mid }$. For each $p\in \pi (G)$, choose a Sylow $p$-subgroup $G_p$ of $G$. The subgroup $\mathrm{\Psi }(G)={\prod }_{p\in \pi (G)}\mathrm{\Phi }(G_p{)}_G$ is called the $\mathit{\text{local}}\mathit{\text{Frattini}}\mathit{\text{subgroup}}$ of $G$. Here, for $J⩽G$, the symbol $J_G$ denotes the core of $J$ in $G$. The subgroup $\mathrm{\Psi }(G)$ is a characteristic subgroup of $G$ that is contained in $\mathrm{\Phi }(G)$, the Frattini subgroup of $G$. We prove that $\mathrm{\Psi }(G)$ satisfies many properties similar to $\mathrm{\Phi }(G)$. Let $J$ be a normal $p$-subgroup of $G$ with $J⩽\mathrm{\Phi }(G)$, and let $S$ be a Sylow $p$-subgroup of $G$. John S. Rose proved that if $J⩽\mathbf{Z}(S)$, then $J⩽\mathrm{\Phi }(S)$. So, in fact, $J⩽\mathrm{\Phi }(S{)}_G⩽\mathrm{\Psi }(G)$. Kaplan, Lev, Mann, and Solomon found various other conditions that would ensure $J$ is contained in $\mathrm{\Phi }(S)$. In this article, we add a few more conditions to this list.