On a classification of irreducible periodic diffeomorphisms on surfaces which commute with certain involution

Ishizaka classified up to conjugacy hyperelliptic periodic automorphisms of a surface. Here, an involution $I$ on a surface $\Sigma_{g}$ is hyperelliptic if and only if $\Sigma_{g}/\langle I \rangle$ is homeomorphic to $S^2$. In this article, we give a classification up to conjugacy for irreducible periodic automorphisms of a surface $\Sigma_{g}$ which commute with involutions $\iota$ such that $\Sigma_{g}/\langle \iota \rangle$ is homeomorphic to $T^{2}$.