On some characterization of general Frattini subgroup of finite soluble group

Let $G$ be a finite soluble group, $\theta$ be a regular subgroup $m$-functor, and $\Phi_\theta(G)$ be the intersection of all maximal $\theta$-subgroups of $G$. Let $n$ be the length of a $G$-series of the group $\mathrm{Soc}(G/\Phi_\theta(G))$, and $k$ be the number of central $G$-chief factors of this series. We prove that in this case $G$ contains $4n-3k$ maximal $\theta$-subgroups whose intersection is $\Phi_\theta(G)$.