On p-supersolubility and solubility of finite groups

Let 𝔉 be a class of groups. A subgroup H of G is said to be weakly 𝔉s-quasinormal in G if G has an S-quasinormal subgroup T such that HT is S-quasinormal in G and (H ∩ T) HG/HG ≤ Z𝔉(G/HG), where Z𝔉(G/HG) is the 𝔉-hypercenter of G/HG. In this paper, we investigate further the influence of weakly 𝔉s-quasinormality of some subgroups on the structure of finite groups. Some new characterizations about p-supersolubility and solubility of finite groups are obtained.