On weakly $s^{\ast}$-permutable subgroups of finite groups

Let $G$ be a finite group. A subgroup $H$ of $G$ is called weakly $s^{\ast}$-permutable in $G$ provided that there exists a subnormal subgroup $T$ of $G$ containing $H_{sG}$ such that $G=HT$ and $|H\cap T:H_{sG}|$ is a square-free integer, where $H_{sG}$ is the largest $s$-permutable subgroup of $G$ contained in $H$. In this paper, we characterize the solvability of $G$ in terms of certain weakly $s^{\ast}$-permutable subgroups.