On a generalisation of finite $T$-groups

Let $\sigma =\{\sigma_i |i\in I\}$ is some partition of all primes $\mathbb{P}$ and $G$ a finite group. A subgroup $H$ of $G$ is said to be $\sigma$-subnormal in $G$ if there exists a subgroup chain $H=H_0\leq H_1\leq \cdots \leq H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is a finite $\sigma_j$-group for some $j \in I$ for $i = 1, \ldots, n$. We call a finite group $G$ a $T_{\sigma}$-group if every $\sigma$-subnormal subgroup is normal in $G$. In this paper, we analyse the structure of the $T_{\sigma}$-groups and give some characterisations of the $T_{\sigma}$-groups.