On complements of coradicals of finite groups

Let $\mathfrak F$ be an $\omega$-local Fitting formation, and $G$ a finite group that can be represented in the form of a product of $n$ subnormal subgroups whose $\mathfrak F$-coradicals are $\omega$-soluble, and whose Sylow $p$-subgroups are abelian for any $p\in\omega$. It is established that there exist $\omega$-complements of the $\mathfrak F$-coradical of $G$. New theorems on the existence of complements of coradicals of a group are obtained as corollaries. For an $\omega$-local formation $\mathfrak F$, conditions are established for the existence of complements and $\omega$-complements of the $\mathfrak F$-coradical of a group in any of its extensions.
Bibliography: 21 titles.