A subgroup $H$ of a finite group $G$ is $s$-permutable in $G$ if $H$ commutes with every Sylow subgroup of $G$. The subgroup $H$ is $s$-semipermutable in $G$ if $PH = HP$ for all Sylow $p$-subgroups $P$ of $G$ such that $(p,{\mid}H{\mid}) = 1$. The subgroup $H$ is weakly $s$-permutable in $G$ (introduced by A.~N. Skiba in [J. Algebra {\bf 315} (2007), no.~1, 192--209; MR2344341 (2008k:20043)]) if there exists a subnormal subgroup $T$ of $G$ such that $G = HT$ and $H \cap T \leq H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all subgroups of $H$ that are $s$-permutable in $G$. Continuing in the $s$-permutable to weakly $s$-permutable line, the subgroup $H$ is weakly $s$-semipermutable in $G$ (introduced by Y. Li et al. in [J. Algebra {\bf 371} (2012), 250--261; 2975395 ]) if there exists a subnormal subgroup $T$ of $G$ and an $s$-semipermutable subgroup $H_{ssG}$ of $G$ contained in $H$ such that $G = HT$ and ${H \cap T \leq H_{ssG}}$. \par In the paper under review, the authors use weakly $s$-semipermutable subgroups of prime power order to study the structure of some finite groups. In particular, they establish conditions for when a group is $p$-nilpotent or $p$-supersolvable. For example, the authors prove the following two results: \par Theorem 3.1. Let $p$ be an odd prime dividing ${\mid}G{\mid}$ and $P$ be a Sylow $p$-subgroup of $G$. If $N_{G}(P)$ is $p$-nilpotent and every minimal subgroup of $P$ is weakly $s$-semipermutable in $G$, then $G$ is $p$-nilpotent. \par Theorem 3.3. Let $G$ be a $p$-solvable group and $p$ be a prime divisor of ${\mid}G{\mid}$. If every maximal subgroup of noncyclic Sylow $p$-subgroup of $F_{p}(G)$ is weakly $s$-semipermutable in $G$, then $G$ is $p$-supersolvable.