Finite Groups with Certain $\mathcal{SSH}$-subgroups

Let $G$ be a finite group. A subgroup $H$ of $G$ is $S$-permutable in $G$ if $H$ permutes with every Sylow subgroup of $G$. A subgroup $H$ of $G$ is called an $\mathcal{SSH}$-subgroup in $G$ if $G$ has an $S$-permutable subgroup $K$ such that $H^{SG} =HK$ and $H^g \cap N_K(H) \leqslant H$, for all $g \in G$, where $H^{SG}$ is the intersection of all $S$-permutable subgroups of $G$ containing $H$. In this paper, we investigate the structure of a finite group $G$ under the assumption that certain subgroups of prime power orders are $\mathcal{SSH}$-subgroups of $G$.