On SS-quasinormalities of the maximal subgroup series of finite groups

Let $G$ be finite group. A subgroup $H$ of $G$ is said to be an $SS$-quasinormal subgroup of $G$, if there exists a subgroup $B$ of $G$ such that $G=HB$ and $H$ permutes with every Sylow subgroup of $B$. Let $\mathrm{\Omega }:G=G_0>G_1>\cdots >G_{n-1}>G_n=1$ be a maximal subgroup series of $G$, where $G_i$ is a maximal subgroup of $G_{i-1}$ for every $i=1,\dots ,n$. In this paper, we investigate the finite groups $G$ that admit an $SS$-quasinormal maximal subgroup series, i.e., all $G_i$ are $SS$-quasinormal in $G$. First, we prove that if $G$ possesses an $SS$-quasinormal maximal subgroup series, then $G$ is solvable. Furthermore, we show that $G$ is supersolvable if and only if $G$ possesses an $SS$-quasinormal maximal subgroup series which is subnormal in $G$.