Intersections of maximal $\theta$-subgroups containing formation radicals

Intersections $\Phi_{\theta, G_{\mathfrak{F}}}$ of all maximal $\theta$-subgroups of a finite group $G$, containing $G_{\mathfrak{F}}$, for nonempty radical formations $\mathfrak{F}$ and the abnormally full subgroup $m_s$-functor $\theta$ are studied.