A contribution to the theory of $\sigma$-properties of a finite group

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 $G\mathrm{/}D$ 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 $F^{\ast }(G)$ 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 $G\mathrm{/}D$ is nilpotent and $D$ avoids every chief factor of $G$ between $V^G$ and $V_G$ for every subnormal subgroup $A$ of $G$.