Finite groups with $C$-quasinormal subgroups

Consider some finite group $G$ and a finite subgroup $H$ of $G$. Say that $H$ is $c$-quasinormal in $G$ if $G$ has a quasinormal subgroup $T$ such that $HT=G$ and $T\cap H$ is quasinormal in $G$. Given a noncyclic Sylow subgroup $P$ of $G$, we fix some subgroup $D$ such that $1<|D|<|P|$ and study the structure of $G$ under the assumption that all subgroups $H$ of $P$ of the same order as $D$, having no supersolvable supplement in $G$, are $c$-quasinormal in $G$.