Pseudo-Anosov action on the $SU(2)$-character variety of $S_2$

In this article, we continue the study of the action of subgroups of the mapping class group on the $SU(2)$-character variety. We prove the existence of a mapping class subgroup on the surface of genus $2$, containing infinitely many pseudo-Anosov elements, which admit an invariant rational function on $SU(2)$-character variety of $S_2$.