Automorphism groups of origami curves

A closed Riemann surface $S$ (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map $\beta :S\to E$ with at most one branch value, where $E$ is a genus one Riemann surface. In this case, $(S,\beta )$ is called an origami pair and $\mathrm{A}\mathrm{u}\mathrm{t}(S,\beta )$ is the group of conformal automorphisms $\varphi$ of $S$ such that $\beta =\beta \circ \varphi$. Let $G$ be a finite group. It is a known fact that $G$ can be realized as a subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(S,\beta )$ for a suitable origami pair $(S,\beta )$. It is also known that $G$ can be realized as a group of conformal automorphisms of a Riemann surface $X$ of genus $g\ge 2$ and with quotient orbifold $X\mathrm{/}G$ also of genus $\gamma \ge 2$. Given a conformal action of $G$ on a surface $X$ as before, we prove that there is an origami pair $(S,\beta )$, where $S$ has genus $g$ and $G\cong \mathrm{A}\mathrm{u}\mathrm{t}(S,\beta )$ such that the actions of ${\rm Aut}(S,\beta)$on$S$andthatof$G$on$X$ are topologically equivalent.