On the weak norm of ${\mathcal{U}}_p$-residuals of all subgroups of a finite group

Let $\mathcal{F}$ be a formation and $G$ a finite group. The weak norm of a subgroup $H$ in $G$ with respect to $\mathcal{F}$ is defined by $N_{\mathcal{F}}(G,H)=N_G(T^{\mathcal{F}})$. In particular, $N_{\mathcal{F}}(G)=N_{\mathcal{F}}(G,G)$. Let $N_{\mathcal{F}}^i(G)$,$i\ge 1$, be a upper series of $G$ by setting $N_{\mathcal{F}}^0(G)=1$, $N_{\mathcal{F}}^{i+1}(G)\mathrm{/}{N}_{\mathcal{F}}^i(G)=N_{\mathcal{F}}(G\mathrm{/}{N}_{\mathcal{F}}^i(G))$ and denoted by $N_{\mathcal{F}}^{\mathrm{\infty }}(G)$ the terminal term of the series. In this paper, for the case $\mathcal{F}\in \{{\mathcal{U}}_p,\mathcal{U}\}$, where ${\mathcal{U}}_p$($\mathcal{U}$,respectively) is the class of all finite $p$-supersolvable groups(supersolvable groups,respectively), we characterize the structure of some given finite groups by the properties of weak norm of some subgroups in $G$ with respect to $\mathcal{F}$. Some of our main results may regard as a continuation of many nice previous work.