{\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}}$.

It remains to show that the resulting action on $S_g$ is determined upto our equivalence in ${\operatorname{Mod}}(S_g)$. Suppose that two permuting $(n,r,k)$-actions $t$ and $t'$ have the same permuting $(n,r,k)$-data set $({\operatorname{{\mathcal D}}},{\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}})$. ${\operatorname{{\mathcal D}}}$ encodes the fixed point data of the periodic transformation $t$, so by a result of J. Nielsen [@N1] (or by a subsequent result of A. Edmonds [@AE Theorem 1.3]), $t$ and $t'$ have to be conjugate by an orientation-preserving homeomorphism $\phi$. Let ${\operatorname{{\mathcal O}}}'$ be the quotient orbifold of the action $t'$, and $\rho': \pi_1^{orb}({\operatorname{{\mathcal O}}}') \to C_n$ be the induced representations. Then $\phi$ induces a map $\varphi_{\#} : \pi_1^{orb}({\operatorname{{\mathcal O}}}) \to \pi_1^{orb}({\operatorname{{\mathcal O}}}')$ such that $\rho'\circ \varphi_{\#} = \rho$ as in [@MK1 Theorem 2.1]. If $\gamma$ is a loop around a cone point in ${\operatorname{{\mathcal O}}}$, then $\rho(\gamma)$ is a loop around a cone point in ${\operatorname{{\mathcal O}}}'$, and these cone points are associated to the same pair in ${\operatorname{{\mathcal D}}}$ since $\rho'(\varphi_{\#}(\gamma)) = \rho(\gamma)$. Hence, $\phi$ maps ${\operatorname{{\mathbb P}}}(t)$ to ${\operatorname{{\mathbb P}}}(t')$ and ${\operatorname{{\mathbb O}}}(t)$ to ${\operatorname{{\mathbb O}}}(t')$ as in Definition \[defn:eq\_perm\_actions\]. Furthermore, ${\operatorname{{\mathbb O}}}_t = {\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}} = {\operatorname{{\mathbb O}}}_{t'}$ by construction, and hence the permuting 