Finite groups with $H_\sigma$-subnormally embedded subgroups

Пусть $G$ be a finite group. Let $\sigma=\{\sigma_i| i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $n$ an integer. We write $\sigma(n)=\{\sigma_i |\sigma_i\cap \pi(n)\ne\varnothing\}$, $\sigma(G)=\sigma(|G|)$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member of $\mathcal{H}\setminus\{1\}$ is a Hall $\sigma_i$-subgroup of $G$ for some $\sigma_i$ and $\mathcal{H}$ contains exact one Hall $\sigma_i$-subgroup of $G$ for every $\sigma_i\in\sigma(G)$. A subgroup $A$ of $G$ is called a $\sigma$-Hall subgroup of $G$ if $\sigma(|A|)\cap\sigma(|G:A|)=\varnothing$. We say that a subgroup $A$ of $G$ is $H_\sigma$-subnormally embedded in $G$ if $A$ is a $\sigma$-Hall subgroup of some $\sigma$-subnormal subgroup of $G$.