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In a previous paper, arXiv:1206.5498, we introduced a new homologicalinvariant $\e$ for the faithful action of a finite group G on an algebraiccurve.  We show here that the moduli space of curves admitting a faithful action of afinite group G with a fixed homological invariant $\e$, if the genus g' of thequotient curve is sufficiently large, is irreducible (and non empty iff theclass satisfies the condition which we define as 'admissibility'). In theunramified case, a similar result had been proven by Dunfield and Thurstonusing the classical invariant in the second homology group of G, H_2(G, \ZZ).  We achieve our result showing that the stable classes are in bijection withthe set of admissible classes $\e$.

Genus stabilization for moduli of curves with symmetries