A normal factorization of a subnormal subgroup of some finite group in connection with local formations and generalized Frattini subgroups. Formation radicals

Let $\pi$ be a set of primes, $\mathfrak{F}=\mathfrak{G}_\pi\mathfrak{F}$ — a local formation of finite groups. The conditions of factorability of a subnormal subgroup $H$ of a finite group $G$ by normal subgroups $H_1$ и $H_2$ such that $H_1\cap H_2=O_\pi(H)$, $H_1\in\mathfrak{F}$, $H_2$ belongs to some generalized Frattini subgroup of a group $G$ and $\pi(H_2/O_\pi(H))\bigcap\pi(\mathfrak{F})=\varnothing$ are investigated. Statements, equivalent to the statements on the respective factorizations, functorially generalized, with the consequences for $\pi=\varnothing$ are achieved. The structure of formation radicals of factorgroups of subnormal subgroups of finite groups in connection with generalized Frattini subgroups is investigated.