Intersections of maximal subgroups in a finite group in connection with the local formations and a generally abnormally full subgroup $m$-functor

Let $\pi$ be a set of primes. The sufficient conditions that must satisfy a local formation $\mathfrak{F}=\mathfrak{G}_{\pi}\mathfrak{F}$, a finite group $G$ and a subgroup $m$-functor $\theta$, under which $\overline{\Delta}_{\pi}^{\mathfrak{F}}(G)=\Phi_{\theta_{\pi}}^{\mathfrak{F}}(G)=\Delta_{\pi}^{\mathfrak{F}}(G)\in\mathfrak{F}$ also $\overline{\Delta}_{\pi,\overline{G_{\mathfrak{F}}}}^{\mathfrak{F}}(G)=\Phi_{\theta_{\pi},\overline{G_{\mathfrak{F}}}}^{\mathfrak{F}}(G)=\Phi_{\theta_{\pi}}^{\mathfrak{F}}(G)=\Delta_{\pi}^{\mathfrak{F}}(G)\subset G_{\mathfrak{F}}\subset G$, if $\mathfrak{F}=\mathfrak{G}_{\pi}\mathfrak{F}$ is radical, are achieved. As the consequences of the main results there were obtained the assertions for $\pi=\varnothing$ and corresponding local formations.