Extremely closed subgroups and a variant on Glauberman's $Z^{\ast }$-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 $⟨H,H^g⟩\cap N_G(H)=H$ for all $g\in G\mathrm{.}$ 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 $G=N_G(H)O_{p^{\mathrm{\prime }}}(G)\mathrm{.}$ This is a variant on the celebrated Glauberman’s $Z^{\ast }$-Theorem.