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 ${\rm Aut}(S,\beta)$ is the group of conformal automorphisms $\phi$ of $S$ such that $\beta=\beta \circ \phi$. Let $G$ be a finite group. It is a known fact that $G$ can be realized as a subgroup of ${\rm Aut}(S,\beta)$ 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 $g \geq 2$ and with quotient orbifold $X/G$ also of genus $\gamma \geq 2$. 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 $G \cong {\rm Aut}(S,\beta)$ such that the actions of ${\rm Aut}(S,\beta)$ on $S$ and that of $G$ on $X$ are topologically equivalent.