finite order homeomorphisms on $S_g({\operatorname{{\mathcal{C}}}})$ induced by $h_s$, for $s = 1,2$. Then $t_{{\operatorname{{\mathcal{C}}}}} = \phi t_{{\operatorname{{\mathcal{C}}}}}\Phi^{-1} = t_{\Phi({\operatorname{{\mathcal{C}}}})}$, so we may assume upto isotopy that $\Phi({\operatorname{{\mathcal{C}}}}) = {\operatorname{{\mathcal{C}}}}$ (as in Remark \[rem:root\_isotopy\]) and that $\Phi(N) = N$. We extend $\Phi\mid_{\widehat{S_g({\operatorname{{\mathcal{C}}}})}}$ to an element $\phi \in {\operatorname{Mod}}(S_g({\operatorname{{\mathcal{C}}}}))$ by coning. Now, $\phi$ maps ${\operatorname{{\mathbb P}}}(t_1)$ to ${\operatorname{{\mathbb P}}}(t_2)$, and ${\operatorname{{\mathbb O}}}(t_1)$ to ${\operatorname{{\mathbb O}}}(t_2)$ bijectively as in Definition \[defn:eq\_perm\_actions\]. Since the $h_s$ and $\Phi$ all preserve $N$, $\phi t_1\phi^{-1}$ is isotopic to $t_2$ preserving ${\operatorname{{\mathbb P}}}(t_2)$ and ${\operatorname{{\mathbb O}}}(t_2)$. Furthermore, for each ${\operatorname{{\mathbb O}}}\in {\operatorname{{\mathbb O}}}(t_1)$, $p(\phi({\operatorname{{\mathbb O}}})) = p({\operatorname{{\mathbb O}}})$. Hence, ${\operatorname{{\mathbb O}}}_{t_1} = {\operatorname{{\mathbb O}}}_{t_2}$ and so $t_1$ and $t_2$ will be equivalent permuting $(n,2r,2k)$-actions.

Conversely, given a permuting $(n,2r,2k)$-action $t$ on $S_g({\operatorname{{\mathcal{C}}}})$ that is nonseparating with respect to ${\operatorname{{\mathcal{C}}}}$, we can reverse the argument to produce the $(r,k)$-permuting root $h$. Let ${\operatorname{{\mathbb P}}}(t) = \{P_1, P_1', \ldots, P_r, P_r'\}$ and ${\operatorname{{\mathbb O}}}(t) = \{{\operatorname{{\mathbb O}}}_1,{\operatorname{{\mathbb O}}}_1', \ldots, {\operatorname{{\mathbb O}}}_k,{\operatorname{{\mathbb O}}}_k'\}$. For $1\leq i\leq r$, we remove disks $D_i$ and $D_i'$ invariant under the action of $t$ around the $P_i$ and $P_i'$ and attaching $r$ annuli to obtain the surface $S_g({\operatorname{{\mathcal{C}}}}\setminus {\operatorname{{\mathcal{C}}}}')$. The condition on the angles $\{\theta_{P_i}(t), \theta_{P_i'}(t)\}$ 