We obtain the complex orbifold structure of the moduli space for one parameter equisymmetric Riemann surfaces of genus two. For each family, by using the orbifold structure, we obtain the points in the moduli corresponding to real algebraic curves and a special form for the period matrices of Riemann surfaces that admit an anticonformal involution. We describe the topological type of anti-conformal involutions admitted by surfaces of the families depending on the type of period matrix.