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 $S\mathrm{/}G$ is at most $2$. As an application, we show that torsion elements in the mapping class group of a surface of genus $\le 2$ are conjugacy distinguished.