On subnormality criteria for subgroups in finite groups

Let H be a subgroup of a finite group G and let S-G(1)(H) be the set of all elements g of G such that H is subnormal in (H, H-g). A result of Wielandt states that H is subnormal in G if and only if G = S-G(1) (H). In this paper, we let A be a subgroup of G contained in S-G(1)(H) and ask if this implies (and therefore is equivalent to) the subnormality of H in (H,A). We show with an example that the answer is no, even for soluble groups with Sylow subgroups of nilpotency class at most 2. However, we prove that the two conditions are equivalent whenever A either is subnormal in G or has p-power index in G (for p any prime number).