A note on $c$-normal subgroups of finite groups

Let $G$ be a finite group. We fix in every non-cyclic Sylow subgroup $P$ of $G$ some its subgroup $D$ satisfying $1<|D|<|P|$ and study the structure of $G$ under assumption that all subgroups $H$ of $P$ with $|H|=|D|$ are $c$-normal in $G$.