On $\sigma$-embedded and $\sigma$-$n$-embedded subgroups of finite groups

Let $G$ be a finite group, and let $\sigma=\{\sigma_i | i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $\sigma(G)=\{\sigma_i | \sigma_i\cap\pi(G)\ne\varnothing\}$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if each nonidentity member of $\mathcal{H}$ is a Hall $\sigma_i$-subgroup of $G$ and $\mathcal{H}$ has exactly one Hall $\sigma_i$-subgroup of $G$ for every $\sigma_i\in \sigma(G)$. A subgroup $H$ of $G$ is said to be $\sigma$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set $\mathcal{H}$ such that $HA^x=A^xH$ for all $A\in\mathcal{H}$ and $G$. A subgroup $H$ of $G$ is said to be $\sigma$-$n$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT = H^G$ and $H\cap T\leqslant H_{\sigma G}$, where $H_{\sigma G}$ is the subgroup of $H$ generated by all those subgroups of $H$ that are $\sigma$-permutable in $G$. A subgroup $H$ of $G$ is said to be $\sigma$-embedded in $G$ if there exists a $\sigma$-permutable subgroup $T$ of $G$ such that $HT = H^{\sigma G}$ and $H\cap H\leqslant H_{\sigma G}$, where $H^{\sigma G}$ is the intersection of all $\sigma$-permutable subgroups of $G$ containing $H$. We study the structure of finite groups under the condition that some given subgroups of $G$ are $\sigma$-embedded and $\sigma$-$n$-embedded. In particular, we give the conditions for a normal subgroup of $G$ to be hypercyclically embedded.