Etale homotopy types of moduli stacks of algebraic curves with symmetries

Using the machinery of etale homotopy theory a’ la Artin-Mazur we determine the etale homotopy types of moduli stacks over $\bar{\text{Q}}$ parametrizing families of algebraic curves of genus g greater than 1 endowed with an action of a finite group G of automorphisms, which comes with a fixed embedding in the mapping class group, 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 the mapping class group.