MVW-extensions of real quaternionic classical groups

Let $G$ be a real quaternionic classical group ${\text{GL}}_n(\text{bH})$, $\text{Sp}(p,q)$ or ${\text{oO}}^{\ast }(2n)$. We define an extension $\breve{G}$ of $G$ with the following property: it contains $G$ as a subgroup of index two, and for every $x\in G$, there is an element $\breve{g}\in \breve{G}\setminus G$ such that $\breve g x\breve{g}^{-1}=x^{-1}$. This is similar to Moeglin-Vigneras-Waldspurger’s extensions of non-quaternionic classical groups.