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

We investigate the properties of the intersection ${\mathrm{I}\mathrm{n}\mathrm{t}}_{\mathfrak{F}}(G)$ of all $\mathfrak{F}$-maximal subgroups of a finite group $G$ for a hereditary formation $\mathfrak{F}$ of finite groups. We prove that ${\mathrm{I}\mathrm{n}\mathrm{t}}_{\mathfrak{F}}(G\mathrm{/}{\mathrm{I}\mathrm{n}\mathrm{t}}_{\mathfrak{F}}(G))\simeq 1$ holds for any finite group $G$ if and only if $\mathfrak{F}$ contains every group $G$ all of whose $\mathfrak{F}$-subgroups are $\mathfrak{F}$-subnormal. As corollaries we obtain the results of A. N. Skiba (2011), J. C. Beidleman and H. Heineken (2011) about ${\mathrm{I}\mathrm{n}\mathrm{t}}_{\mathfrak{F}}(G)$ for a hereditary saturated formation $\mathfrak{F}$.