Let's work over the complex numbers $\mathbb{C}$. Let $g\geq3$ be an integer. Let $\mathcal{M}_g$ be moduli stack of smooth genus $g$ curves. Let $M_g$ be the corresponding coarse moduli scheme. They share an open subscheme $M_g^\circ$ parametrizing automorphism-free smooth genus $g$ curves.

What is the fundamental group $\pi_1(M_g^\circ)$? How does it compare to the following notions:

(1) The orbifold mapping class group $\pi_1(\mathcal{M}_g)$,

(2) The topological fundamental group $\pi_1(M_g)$,

(3) The mapping class group of a genus $g$ surface, $\mathrm{Mod}_g$?