A congruence subgroup property for symmetric mapping class groups

We prove the congruence subgroup property for the centralizer of a finite subgroup G in the mapping class group of a hyperbolic oriented and connected surface of finite topological type S such that the genus of the quotient surface S / G is at most 2. As an application, we show that torsion elements in the mapping class group of a surface of genus ≤ 2 ≤ 2 are conjugacy distinguished.