A new characterization of finite $\sigma$-soluble $P\sigma T$-groups

Let $\sigma =\{\sigma_{i} \mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be $\sigma$-soluble if every chief factor $H/K$ of $G$ is a $\sigma_{i}$-group for some $i=i(H/K)$. A set ${\mathcal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every member $\ne 1$ of ${\mathcal H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in \sigma $ and ${\mathcal H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $i$ such that $\sigma_{i}\cap \pi (G)\ne \varnothing$. A subgroup $A$ of $G$ is said to be ${\sigma}$-quasinormal or ${\sigma}$-permutable in $G$ if $G$ has a complete Hall $\sigma$-set $\mathcal H$ such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \mathcal H$. We obtain a new characterization of finite $\sigma$-soluble groups $G$ in which $\sigma$-permutability is a transitive relation in $G$.