  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 $G$.
