Finite groups with normally embedded subgroups

A subgroup H of the finite group G is said to be quasinormally (resp. S -quasinormally) embedded in G if for every Sylow subgroup P of H , there is a quasinormal (resp. S -quasinormal) subgroup K in G such that P is also a Sylow subgroup of K . Groups with certain quasinormally (resp. S -quasinormally) embedded subgroups of prime-power order are studied. For example, if a group G has a normal subgroup H such that G / H ∈ ℱ and such that for each Sylow subgroup P of H , every member in some ℳ d ( P ) is quasinormally embedded in G , then G ∈ ℱ: here ℳ d ( P ) is a set of maximal subgroups of P with intersection the Frattini subgroup.