On the existence of complements for residuals of finite groups

L.A. Shemetkov's theorem on the complementability of the $\mathfrak{F}$-residual of a finite group is developed in the article. For a local Fitting formation $\mathfrak{F}$, it is proved that, if a group $G$ is representable in the form $G=AB$, where $A$ and $B$ are subnormal subgroups of $G$, the subgroups $A^\mathfrak{F}$ and $B^\mathfrak{F}$ are $\pi(\mathfrak{F})$-solvable and normal in $G$, and Sylow $p$-subgroups of $A^\mathfrak{F}$ and $B^\mathfrak{F}$ are abelian for every $p \in \pi(\mathfrak{F})$, then every $\mathfrak{F}$-normalizer of $G$ is the complement for an $\mathfrak{F}$-residual of $G$.