Nonequivalent families of group actions on Riemann surfaces

We produce for each natural number n≥3 two 1-parameter families of Riemann surfaces admitting automorphism groups with two cyclic subgroups H1 and H2 of order 2n, which are conjugate in the group of orientation-preserving homeomorphisms of the corresponding Riemann surfaces, but not conjugate in the group of conformal automorphisms.This property implies that the subvariety Mg(H1) of the moduli space Mg consisting of the points representing the Riemann surfaces of genus g admitting a group of automorphisms topologically conjugate to H1 (equivalently to H2) is not a normal subvariety.