Extremely closed subgroups and a variant on Glauberman's Z^*-theorem

Let G be a finite group and let H be a subgroup of G. We say that H is extremely closed in G if H,Hᵍ∩ NG(H)=H for all g∈ G. In this paper, we determine the structure of finite groups with an extremely closed abelian p-subgroup for some prime p. In particular, we show that if G contains such a subgroup H, then G=NG(H)Op'(G). This is a variant on the celebrated Glauberman's Z^*-Theorem.