Some special families of hyperelliptic curves

Let ${\text{L}}_g^G$ denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces ${\text{L}}_g^G$ for $G\text{iso}{\mathbb{Z}}_n,{\mathbb{Z}}_2\text{o}{\mathbb{Z}}_n,{\mathbb{Z}}_2\text{o}{A}_4$, or $S{L}_2(3)$. We show that for $G\text{iso}{\mathbb{Z}}_n,{\mathbb{Z}}_2\text{o}{\mathbb{Z}}_n$, the space ${\text{L}}_g^G$ is a rational variety and find generators of its function field. For $G\text{iso}{\mathbb{Z}}_2\text{o}{A}_4,S{L}_2(3)$ we find a necessary condition in terms of the coefficients, whether or not the curve belongs to ${\text{L}}_g^G$. Further, we describe algebraically the loci of such curves for $g\le 12$ and show that for all curves in these loci the field of moduli is a field of definition.