On NR*-subgroups of finite groups

Let G be a finite group and let H be a subgroup of G . H is said to be an N R*-subgroup of G if there exists a normal subgroup T of G such that G = HT and if whenever K 1hd H and g ∈ G, then K^g ∩ H ∩ T ≤ K. A number of new characterizations of a group G are given, under the assumption that all Sylow subgroups of certain subgroups of G are N R*-subgroups.