The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces

Denote the set of all holomorphic mappings of a genus 3 Riemann surface $S_3$ onto a genus 2 Riemann surface $S_2$ by $\operatorname{Hol}(S_3,S_2)$. Call two mappings $f$ and $g$ in $\operatorname{Hol}(S_3,S_2)$ equivalent whenever there exist conformal automorphisms $\alpha$ and $\beta$ of $S_3$ and $S_2$ respectively with $f\circ\alpha=\beta\circ g$. It is known that $\operatorname{Hol}(S_3,S_2)$ always consists of at most two equivalence classes.
We obtain the following results: If $\operatorname{Hol}(S_3,S_2)$ consists of two equivalence classes then both $S_3$ and $S_2$ can be defined by real algebraic equations; furthermore, for every pair of inequivalent mappings $f$ and $g$ in $\operatorname{Hol}(S_3,S_2)$ there exist anticonformal automorphisms $\alpha^-$ and $\beta^-$ with $f\circ\alpha^-=\beta^-\circ g$. Up to conformal equivalence, there exist exactly three pairs of Riemann surfaces $(S_3,S_2)$ such that $\operatorname{Hol}(S_3,S_2)$ consists of two equivalence classes.