Finite group with Hall normally embedded minimal 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 H G , where H G is the normal closure of H in G, that is, the smallest normal subgroup of G containing H. A group G is called an HNE2-group if all cyclic subgroups of order 2 and 4 of G are Hall normally embedded in G. In this paper, we prove that all HNE2-groups are 2-nilpotent. Furthermore, we also characterize the structure of finite group all of whose maximal subgroups are HNE2-groups. Finally, we determine finite non-solvable groups all of whose second maximal subgroups are HNE2-groups.