Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$.

Let $H_{X,d}$ be the Hilbert scheme of smooth hypersurfaces of degree $d$ in $X$.

Note that $H_{X,d}$ is a quasi-projective scheme over $\mathbb C$ equipped with an action of the automorphism group $G:=\mathrm{Aut}(X)$ of $X$.

Is the action of $G$ on $H_{X,d}$ proper? That is, is the DM-stack $[G\backslash H_{X,d}]$ separated?

There is a natural representable morphism of stacks

$$[G\backslash H_{X,d}] \to \mathcal M_g$$

What are some of its abstract properties? Is it quasi-finite? Does it induce an open immersion on coarse spaces?