On the intersection of the $\mathcal{F}$-maximal subgroups and the generalized $\mathcal{F}$-hypercentre of a finite group

Let $\mathcal{F}$ be a class of groups. A chief factor $H\mathrm{/}K$ of a group $G$ is called \emph{$\mathcal{F}$-central in $G$} provided $(H/K)\rtimes (G/C_{G}(H/K)) \in {\cal F}$\mathrm{.}Wewrite$Z_{\pi{\cal F}}(G)$ to denote the product of all normal subgroups of $G$ whose $G$-chief factors of order divisible by at least one prime in $\pi$ are $\mathcal{F}$-central. We call $Z_{\pi \mathcal{F}}(G)$ the $\pi \mathcal{F}$-hypercentre of $G$. A subgroup $U$ of a group $G$ is called \emph{$\mathcal{F}$-maximal} in $G$ provided that (a) $U\in \mathcal{F}$, and (b) if $U\le V\le G$ and $V\in \mathcal{F}$, then $U=V$. In this paper we study the properties of the intersection of all $\cal F$-maximal subgroups of a finite group. In particular, we analyze the condition under which $Z_{\pi \mathcal{F}}(G)$ coincides with the intersection of all $\cal F$-maximalsubgroupsof$G$.