Is every spin group $Spin(n,R)$ over the reals contained in some metaplectic group $Mp(m,R)$ for some $m$ in such a way that the spin representation is obtained by restriction of the metaplectic representation?



Is the spin group in a metaplectic group?
Let $\alpha : 1\to K \to Spin(m)\to SO(m)\to 1$ be the universal central extension, so that $K=\mathbf Z$ if $m= 2$ and $K=\mathbf Z/2$ if $m\geq 3$.


Let $\beta : 1\to \mathbf Z \to \tilde{Sp}(n)\to {Sp}(n)\to 1$ be the universal central extension.


Let $\gamma : 1\to \mathbf Z/2 \to {Mp}(n)\to {Sp}(n)\to 1$ be the metaplectic central extension.


I interpret the question as : can one find a morphism $f:SO(m)\to Sp(n)$ such that the pullback $f^\*(\gamma)$ of $\gamma$ along $f$ be isomorphic to $\alpha$ ?


It seems that the answer is always no. Indeed, 


* for $m\geq 3$, $f^\*(\gamma)$ is the image of $f^\*(\beta)$, which is an extension of $SO$ by $\mathbf Z$ and hence is trivial.
* for $m=2$, it is impossible since $\mathbf Z\neq \mathbf Z/2$. Nevertheless, in this case, if $f$ is the inclusion $SO(2)\to SL(2)$, then $f^\*(\beta)$ is isomorphic to $\alpha$.


