**MSC:**- 14F35 Homotopy theory; fundamental groups, See also {14E20,
- 14H10 Families, moduli (algebraic)

Artin-Mazur we determine the etale homotopy types of moduli

stacks over $\overline{\Bbb{Q}}$ parametrizing families of

algebraic curves of genus $g \geq 2$ endowed with an action

of a finite group $G$ of automorphisms, which comes with a

fixed embedding in the mapping class group $\Gamma_g$, such

that in the associated complex analytic situation the action

of $G$ is precisely the differentiable action induced by

this specified embedding of $G$ in $\Gamma_g$.