On p-nilpotency of finite groups with some subgroups π-quasinormally embedded

A subgroup H of a group G is said to be pi-quasinormal in G if it permutes with every Sylow subgroup of G, and H is said to be pi-quasinormally embedded in G if for each prime dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some pi-quasinormal subgroups of G. We characterize p-nilpotentcy of finite groups with the assumption that some maximal subgroups, 2-maximal subgroups, minimal subgroups and 2-minimal subgroups are pi-quasinormally embedded, respectively.