On HC-subgroups of a finite group

A subgroup $H$ of a finite group $G$ is said to be an $\mathscr{H}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g \cap N_T(H)\leq H$ for all $g\in G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that certain subgroups of $G$ of arbitrary prime power order are $\mathscr{H}C$-subgroups of $G$.