On the normalizers of FF-residuals of all subgroups of a finite group

Let FF be a non-empty formation and G   be a finite group. We define the subgroup NF(G)NF(G) to be the intersection of the normalizers of the FF-residuals of all subgroups of G  . Set NF0(G)=1 and define NFi+1(G)/NFi(G)=NF(G/NFi(G)) for i⩾1i⩾1. By NF∞(G) denote the terminal term of this ascending series. Let F=NFF=NF be the class of all groups whose FF-residuals are nilpotent. In this paper, we release several deep relationships between G=NF∞(G) and G∈FG∈F. We characterize the formations FF such that G∈FG∈F would imply that G=NF∞(G), and the formations FF such that G=NF∞(G) would imply that G∈FG∈F. We also study the relationship between NF∞(G) and IntF(G)IntF(G), the intersection of all F-maximal subgroups of G  . We characterize the formations FF such that NF∞(G) is contained in IntF(G)IntF(G) for each group G  , and the formations FF such that NF∞(G) coincides with IntF(G)IntF(G) for each group G  . Moreover, for an arbitrary non-empty formation FF, the properties of NF∞(G) are investigated provided that G is (p-)soluble. Some recent results are generalized.