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

Let $G$ be a finite group and $\mathfrak F$ a formation of finite groups. We say that a subgroup $H$ of $G$ is $\mathfrak F_h$-normal in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is a normal Hall subgroup of $G$ and $(H \cap T)H_G/H_G$ is contained in the $\mathfrak F$-hypercenter $Z_\propto^\mathfrak F(G/H_G)$ of $G/H_G$. In this paper, we obtain some results about the $\mathfrak F_h$-normal subgroups and use them to study the structure of finite groups.