On maximal abnormal subgroups of finite groups

A subgroup $m$-functor $\Theta$ is a function that maps each group $G$ to some set $\Theta(G)$ consisting of maximal subgroups of $G$ and the group $G$ itself; it is assumed that $\Theta(G^\alpha)=(\Theta(G))^\alpha$ for any automorphism $\alpha$ of $G$. We establish the structure of the functor generalized Frattini subgroup and its influence on the properties of the group.