Let $\mathfrak{ J}$ be a class of non-abelian simple groups and $\mathfrak{ X}$ be a class of groups. A chief factor $H/K$ of a group $G$ is called $\mathfrak{ X}$-central in $G$ provided $(H/K)\rtimes G/C_G(H/K)\in\mathfrak{ X}$. We say that $G$ is a $\mathfrak{ J}cs$-$\mathfrak{ X}$-\emph{ group} if every chief $\mathfrak{ X}$-factor of $G$ is $\mathfrak{ X}$-central and other chief factors of $G$ are simple $\mathfrak{ J}$-groups. We use $\mathfrak{ X}_{ \mathfrak{ J}cs}$ to denote the class of all $\mathfrak{ J}cs$-$\mathfrak{ X}$-groups. A subgroup $U$ of a group $G$ is called $\mathfrak{ X}$-\emph{ maximal} in $G$ provided that $(a)$ $U\in\mathfrak{ X}$, and $(b)$ if $U\leq V \leq G$ and $V\in\mathfrak{ X}$, then $U = V$. In this paper we described the structure of $\mathfrak{ J}cs$-$\mathfrak{ H}$-groups for a solubly saturated formation $\mathfrak{ H}$ and all hereditary saturated formations $\mathfrak{ F}$ containing all nilpotent groups such that the $\mathfrak{ F}_{ \mathfrak{ J}cs}$-hypercenter of $G$ coincides with the intersection of all $\mathfrak{ F}_{ \mathfrak{ J}cs}$-maximal subgroups of $G$ for every group $G$.
Categories: math.GR