  Using the machinery of etale homotopy theory a' la Artin-Mazur we determine
the etale homotopy types of moduli stacks over $\bar{\Q}$ 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.
