On $s$-semipermutable and weakly $s$-permutable subgroups

Let $H$ be a subgroup of a finite group $G$. $H$ is said to be $s$-semipermutable in $G$ if $HG_{p}=G_{p}H$ for any Sylow $p$-subgroup $G_{p}$ of $G$ with $(p,|H|)=1$; $H$ is called weakly $s$-permutable in $G$ 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 those subgroups of $H$ which are $s$-permutable in $G$. We fix in every non-cyclic Sylow subgroup $P$ of $G$ a subgroup $D$ with $1<|D|<|P|$ and study the structure of $G$ under the assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is either $s$-semipermutable or weakly $s$-permutable in $G$. Some recent results are generalized and unified.