On $\mathfrak F_n$-normal subgroups of finite groups

Given a class $\mathfrak F$ of finite groups, a subgroup $H$ of a group $G$ is called $\mathfrak F_n$-normal in $G$, if there exists a normal subgroup $T$ of $G$ such that $HT$ is a normal subgroup of $G$ and $(H\cap T)H_G/H_G$ is contained in the $\mathfrak F$-hypercenter $Z^\mathfrak F_\infty(G/H_G)$ of $G/H_G$. We obtain some results about the $\mathfrak F_n$-normal subgroups and use them to study the structure of some groups.