  Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let
${\cal L}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with
$A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is
$\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K)) \in\mathfrak{F}$.
  We study the structure of $G$ under the hypothesis that every chief factor
  of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every
subgroup $A\in {\cal L}_{\mathfrak{F}}(G)$. As an application, we prove that a
finite soluble group $G$ is a $PST$-group if and only if $A^{G}/A_{G}\leq
Z_{\infty}(G/A_{G})$ for every subgroup $A\in {\cal L}_{\mathfrak{N}}(G)$,
where $\mathfrak{N}$ is the class of all nilpotent groups.
