The locus of curves with prescribed automorphism group

Let G be a finite group, and $g\ge 2$. We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g=3 (including equations for the corresponding curves), and for $g\le 10$ we classify those loci corresponding to “large” G.