Coble's restriction-map problem for automorphisms of Coble surfaces

Let $X$ be a Coble surface with Coble curve $C\simeq\mathbb{P}^1$, and let

$$\rho:\operatorname{Aut}(X)\to\operatorname{Aut}(C)\simeq\operatorname{PGL}(2)$$

be the restriction homomorphism. Coble's restriction-map problem. Is $\ker(\rho)$ trivial for a general Coble surface $X$, and what is the image

$$\operatorname{Im}(\rho)\subset\operatorname{PGL}(2)?$$

This is the source's explicit formulation of Coble's question; the supplied text gives no resolution for either the kernel or image problem.