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

We prove that $G$ is a finite $\sigma$-soluble group with transitive $\sigma$-permutability if and only if the following hold: (i) $G$ possesses a complete Hall $\sigma$-set $\mathcal{H}=\{H_{1}, \dots , H_{t}\}$ and a normal subgroup $N$ with $\sigma$-nilpotent quotient $G/N$ such that $H_{i}\cap N\leq Z_{\mathfrak{U}}(H_{i})$ for all $i$; and (ii) every $\sigma _{i}$-subgroup of $G$ is $\tau_{\sigma}$-permutable in $G$ for all $\sigma _{i}\in \sigma (N)$.