A contribution to the theory of σ-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 $G/D$ is supersoluble and D avoids every chief factor of G between VG and VG 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 VG and VG for every subnormal subgroup A of G.