Finite groups with 2-minimal or 2-maximal subgroups are Hall normally embedded subgroups

Let G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure H-G of H in G. In this paper, we investigate the structure of a finite group G under the assumption that certain 2-minimal subgroups and 2-maximal subgroups are Hall normally embedded in G, respectively. Some conditions for a finite group to be p-nilpotent and supersolvable are given.