MVW-extensions of real quaternionic classical groups

Let $G$ be a real quaternionic classical group $\GL_n(\bH)$, $\Sp(p,q)$ or $\oO^*(2n)$. 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.

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