On the intersection of all maximal $\mathfrak F$-subgroups of a finite group

Let $\mathfrak F$ be a class of groups. A subgroup $H$ of a group $G$ is said to be a maximal $\mathfrak F$-subgroup of $G$ if $H \in \mathfrak F$ and has no a subgroup $E \in \mathfrak F$ such that $H \le E$. The symbol $\Sigma_{\mathfrak F}(G)$ denotes the intersection of all maximal $\mathfrak F$-subgroups of $G$. We study the influence of the subgroup $\Sigma_{\mathfrak F}(G)$ on the structure of $G$.