Finite groups with given weakly $\sigma$-permutable subgroups

Let $G$ be a finite group and let $\sigma=\{\sigma_i\mid i\in I\}$ be a partition of the set of all primes $\mathbb P$. A set $\mathscr H$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if each nonidentity member of $\mathscr H$ is a Hall $\sigma_i$-subgroup of $G$ and $\mathscr H$ 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 $\mathscr H$ such that $HA^x=A^xH$ for all $A\in\mathscr H$ and all $x\in G$. A subgroup $H$ of $G$ is said to be weakly $\sigma$-permutable in $G$ if there exists a $\sigma$-subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq H_{\sigma G}$, where $H_{\sigma G}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $\sigma$-permutable in $G$. We study the structure of $G$ under the condition that some given subgroups of $G$ are weakly $\sigma$-permutable in $G$. In particular, we give the conditions under which a normal subgroup of $G$ is hypercyclically embedded. Some available results are generalized.