Let $C$ be a smooth genus two hyperelliptic curve and $\mathcal{M}_C$ be a moduli space of stable rank two vector bundles of fixed degree(or fixed determinant). Then is $\mathrm{Aut}(\mathcal{M}_C)\subset\mathrm{Aut}(C)?$. The motivation is following. Let $Y$ be an index two prime Fano threefold of degree $4$, one can show that such threefold is isomorphic to a moduli space of stable vector bundle of rank two fixed determinant over a genus two curve $C$ such that its degree is odd(say degree=1). Then, I am wondering if $\mathrm{Aut}(Y)\subset\mathrm{Aut}(C)?$. In the paper Hilbert scheme of lines and conics and automorphism groups of Fano threefolds, the authors show that $\mathrm{Aut}(Y)\subset\mathrm{Aut}(J(C))$. But I think $\mathrm{Aut}(J(C))$ is not contained in $\mathrm{Aut}(C)$ since $J(C)$ has some translation which is not coming from $\mathrm{Aut}(C)$. On the other hand, for index one degree 12 prime Fano threefold $X$. The authors show that $\mathrm{Aut}(X)\subset\mathrm{Aut}(C)$, in this case $X$ is also purely determined by $C$.
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The moduli space of rank 2 vector bundles on $C$ with fixed determinant of odd degree is a smooth complete intersection of two quadrics in $\ \mathbb{P}^5\qquad$ (P. Newstead, Topology 7 (1968), 205-215); in an appropriate system of coordinates, it is given by $\sum X_i^2=\sum \lambda _iX_i^2=0$, for $\lambda_0,\ldots ,\lambda_5\in\mathbb{C}$ distinct. It always admits a group $(\mathbb{Z}/2)^5$ of automorphisms (acting by changing the sign of the coordinates). On the other hand, $\operatorname{Aut}(C)=\mathbb{Z}/2 $ for $C$ general.
