On ss-quasinormal or weakly s-permutably embedded subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be an ss-quasinormal subgroup of G if there is a subgroup B of G such that (G=HB) and H permutes with every Sylow subgroup of B; H is said to be weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup (H_{se}) of G contained in H such that (G=HT) and (Hcap Tle H_{se}). We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying (1<|D|<|P|) and study the structure of G under the assumption that every subgroup H of P with (|H|=|D|) is either ss-quasinormal or weakly s-permutably embedded in G. Some recent results are generalized and unified.