On the $\mathcal{F}^*$-norm of a finite group

Let $G$ be a finite group and $\mathcal{F}$ be a non-empty formation. We define the $\mathcal{F}^$-norm, denoted by $N\_{\mathcal{F}}^{}(G)$, to be intersection of the normalizers of the $\mathcal{F}$-residuals of all $F$-subgroups of $G$, where $F=\mathcal{N}\mathcal{F}$ is the class of all groups whose $\mathcal{F}$-residuals are nilpotent. In this paper, we research the properties of $N\_{\mathcal{F}}^{}(G)$ and investigate the relationship between $N\_{\mathcal{F}}^{}(G)$ and $N\_{\mathcal{F}}(G),$ where $N\_{\mathcal{F}}(G)$ is the intersection of the normalizers of the $\mathcal{F}$-residuals of all subgroups of $G.$ We show that $N\_{\mathcal{F}}^{\*}(G)=N\_{\mathcal{F}}(G)$ if $\mathcal{A}\subseteq \mathcal{F}\subseteq\mathcal{N}$.