We will work over complex number field $\mathbb{C}$. Let $\mathscr{M}_h$ be the moduli functor for canonically polarized manifolds with $h$ the Hilbert polynomial. Let us denote by $M_h$ the coarse moduli space of $\mathcal{M}_h$. For any $X\in \mathscr{M}_h({\rm Spec}(\mathbb{C}))$, we know that its Kuranishi family $f:\mathscr{X}\to S$ exists, with $f^{-1}(s_0)\simeq X$. After shrinking $S$, we may assume that $f:\mathscr{X}\to S\in \mathscr{M}_h(S)$. Since $H^0(X,T_X)=0$, the Kuranishi family $f:\mathscr{X}\to S$ is universal. Moreover, the finite automorphism group $\mathrm{Aut}(X)$ acts on $S$ with $s_0$ the fixed point due to the definition of Kuranishi family.

My question is: can $S/\mathrm{Aut}(X)$ be seen as a neighborhood of $[X]\in M_h$? In other words, $M_h$ can be obtained by glueing together the quotient of Kuranishi spaces.