Finite Groups with ℋ-Subgroups

Let be a saturated formation containing , and G be a finite group. Li etc. proposed a problem: whether there is a normality such that the following two statements are equivalent: (i) . (ii) There exists a normal subgroup H of G such that and for each Sylow subgroup P of F*(H), every member in some satisfies the above normality in G. In this paper, we find a normality satisfying the above problem. Moreover, by using the concept of ℋ-subgroups, we obtain other results about the influence of the members of some fixed on the structure of G.