On one generalization of finite $\sigma$-nilpotent groups

Let $G$ be a finite group. Let $\sigma=\{\sigma_i\mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $n$ an integer. Let $\sigma(n)=\{\sigma_i\mid\sigma_i\cap\pi(n)\ne\varnothing\}$, $\sigma(G)=\sigma(|G|)$. A set $l\in\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member of $\mathcal{H}\setminus\{l\}$ 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 $\sigma_i\in\sigma(G)$. If $G$ possesses a complete Hall $\sigma$-set, then it is said to be $\sigma$-full. A subgroup $A$ of $G$ is called: (i) a $\sigma$-Hall subgroup of $G$ if $\sigma(A)\cap\sigma(|G:A|)=\varnothing$; (ii) $H_\sigma$-normally embedded in $G$ if $A$ is a $\sigma$-Hall subgroup of some normal subgroup of $G$. In this paper, we study $\sigma$-full groups $G$ whose all subgroups are $H_\sigma$-normally embedded in $G$.