Finite group with given 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 \(H^G\) $H^G$ . In this paper, we fix a subgroup D of Sylow p-subgroup P of G with \(1<|D|<|O_p(G)|\) $1<|D|<|O_p(G)|$ and study the structure of G under the assumption that all subgroups H of P with order \(|H|=|D|\) $|H|=|D|$ are Hall normally embedded in G.