Equivalence of Group Actions on Riemann Surfaces

We produce for each natural number $n\ge 3$ two 1–parameter families of Riemann surfaces admitting automorphism groups with two cyclic subgroups $H_1$ and $H_2$ of orden $2^n$, that are conjugate in the group of orientation–preserving homeomorphism of the corresponding Riemann surfaces, but not conjugate in the group of conformal automorphisms. This property implies that the subvariety ${\mathcal{M}}_g(H_1)$ of the moduli space ${\mathcal{M}}_g$ consisting of the points representing the Riemann surfaces of genus $g$ admitting a group of automorphisms topologically conjugate to $H_1$ (equivalently to $H_2$) is not a normal subvariety.