On finite $P\sigma T$-groups

Let $\sigma =\{{\sigma }_i\mathrm{\mid }i\in I\}$ be some partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be \emph{$\sigma$-soluble} if every chief factor $H\mathrm{/}K$ of $G$ is a ${\sigma }_i$-group for some $i=i(H\mathrm{/}K)$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma$-set} of $G$ if every member $\mathrm{\ne }1$ of $\mathcal{H}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some $\sigma_{i}\in \sigma $and${\cal H}$containsexactoneHall$\sigma_{i}$-subgroupof$G$forevery$i \in I$suchthat$\sigma_{i}\cap \pi (G)\ne \emptyset$\mathrm{.}Asubgroup$A$of$Gis said to be \emph{{\sigma}$-permutable} or \emph{$\sigma$-quasinormal} in $G$ if $G$ has a complete Hall $\sigma$-set $\mathcal{H}$ such that $A{H}^x=H^{x}A$ for all $x\in G$ and all $H\in \mathcal{H}$. We obtain a characterization of finite $\sigma$-soluble groups $G$ in which $\sigma$-quasinormality is a transitive relation in $G$.