On Φ-τ-quasinormal subgroups of finite groups

‎Let τ be a subgroup functor and H a p-subgroup of a finite group G‎. ‎Let G¯=G/HG and H¯=H/HG‎. ‎We say that H is Φ-τ-quasinormal in G if for some S-quasinormal subgroup T¯ of G¯ and some τ-subgroup S¯ of G¯ contained in H¯‎, ‎H¯T¯ is S-quasinormal in G¯ and H¯∩T¯≤S¯Φ(H¯)‎. ‎In this paper‎, ‎we study the structure of a group G under the condition that some primary subgroups of G are Φ-τ-quasinormal in G‎. ‎Some new characterizations about p-nilpotency and solubility of finite groups are obtained.