On the intersections of maximal subgroups of finite groups containing formation radicals

For nonempty radical formation $\mathfrak{F}$ and a finite group $G$ the following statement was proved: if there exist maximal subgroups of $G$ containing $G_{\mathfrak{F}}$, but not containing $G_{\mathfrak{FN}}$, that is $\Phi_{G_{\mathfrak{F}},\overline{G_{\mathfrak{FN}}}}(G)\ne G$, and the factor group $\tilde{\mathrm{F}}_{\Phi_{G_{\mathfrak{F}}}}(G)\cap \Phi_{G_{\mathfrak{F}},\overline{G_{\mathfrak{FN}}}}(G)/\Phi_{G_{\mathfrak{F}}}(G)$ is solvable, then $\Phi_{G_{\mathfrak{F}}}(G)=\Phi_{G_{\mathfrak{F}},\overline{G_{\mathfrak{FN}}}}(G)\subset G_{\mathfrak{FN}}\subseteq\mathrm{F}_{\Phi_{G_{\mathfrak{F}}}}(G)$. In particular, if $G\ne G_{\mathfrak{F}}$ and $\mathrm{Soc}(G/\Phi_{G_{\mathfrak{F}}}(G))=\tilde{\mathrm{F}}_{\Phi_{G_{\mathfrak{F}}}}(G)/\Phi_{G_{\mathfrak{F}}}(G)$ is solvable, then $\Phi_{G_{\mathfrak{F}}}(G)=\Phi_{G_{\mathfrak{F}},\overline{G_{\mathfrak{FN}}}}(G)\subset G_{\mathfrak{FN}}=\tilde{\mathrm{F}}_{\Phi_{G_{\mathfrak{F}}}}(G)$. The corresponding consequences were obtained for products of non-empty radical formations, in particular for $\mathfrak{F}=\mathfrak{N}^{n-1}$, $n$ is any natural number.