  We characterize some classes of finite soluble groups. In particular, we
prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal
subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor
of $G$ between $V^{G}$ and $V_{G}$ for every maximal subgroup $V$ of the
generalized Fitting subgroup $F^{*}(G)$ of $G$; a finite soluble group $G$ is a
$PST$-group (that is, Sylow permutability is a transitive relation on $G$) if
and only if $G$ has a normal subgroup $D$ such that $G/D$ is nilpotent and $D$
avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every
subnormal subgroup $A$ of $G$.
