Finite groups with a system of generalized central elements

Let $H$ be a normal subgroup of a finite group $G$. A number of authors have investigated the structure of $G$ under the assumption that all minimal or maximal subgroups in Sylow subgroups of $H$ are well-situated in $G$. A general approach to the results of that kind is proposed in this article. The author has found the conditions for $p$-elements of $H$ under which $G$-chief $p$-factors of $H$ are $\mathfrak{F}$-central in $G$.