(ii) there exists $k$ mutually disjoint pairs $\{{\operatorname{{\mathbb O}}}_i,{\operatorname{{\mathbb O}}}_i'\}$ of distinguished nontrivial orbits in ${\operatorname{{\mathbb O}}}_t$ such that ${\operatorname{{\mathbb O}}}_i\sim {\operatorname{{\mathbb O}}}_i'$, for $1 \leq i \leq k$, and

(iii) $S({\operatorname{{\mathbb P}}}_{r,k}) = S(t)$.

\[thm:actions-nonseproots\] Let ${\operatorname{{\mathcal{C}}}}$ be a nonseparating multicurve in $S_g$. Then for $n \geq 1$, equivalence classes of permuting $(n,2r,2k)$-actions on $S_g({\operatorname{{\mathcal{C}}}})$ that are nonseparating with respect to ${\operatorname{{\mathcal{C}}}}$ correspond to the conjugacy classes in ${\operatorname{Mod}}(S_g)$ of $(r,k)$-permuting roots of $t_{{\operatorname{{\mathcal{C}}}}}$ of degree $n$.

First, we shall prove that a conjugacy class of an $(r,k)$-permuting root $h$ of $t_{{\operatorname{{\mathcal{C}}}}}$ of degree $n$ yields an equivalence class of a permuting $(n,2r,2k)$-action that is nonseparating with respect to ${\operatorname{{\mathcal{C}}}}$. We assume without loss of generality that $r,k > 0$, with the implicit understanding that, when either of them is zero, the corresponding arguments may be disregarded.

Let ${\operatorname{{\mathcal{C}}}}_{r,k}(h) = \{{\operatorname{{\mathcal{C}}}}_1', {\operatorname{{\mathcal{C}}}}_2',\ldots, {\operatorname{{\mathcal{C}}}}_r', {\operatorname{{\mathcal{C}}}}_1,{\operatorname{{\mathcal{C}}}}_2,\ldots, {\operatorname{{\mathcal{C}}}}_k\}$ be the $(r,k)$-partition of ${\operatorname{{\mathcal{C}}}}$ associated with $h$. Choose a closed tubular neighborhood $N$ of ${\operatorname{{\mathcal{C}}}}$, and consider $S_g({\operatorname{{\mathcal{C}}}})$ as in Definition \[defn:sgc\]. By isotopy, we may assume that $t_{{\operatorname{{\mathcal{C}}}}}({\operatorname{{\mathcal{C}}}})={\operatorname{{\mathcal{C}}}}$, $t_{{\operatorname{{\mathcal{C}}}}}(N)=N$, and $t_{{\operatorname{{\mathcal{C}}}}}\vert_{\widehat{S_g({\operatorname{{\mathcal{C}}}})}}=\text{id}_{\widehat{S_g({\operatorname{{\mathcal{C}}}})}}$. Suppose that $h$ is a root of $t_{{\operatorname{{\mathcal{C}}}}}$ of degree $n$, then by Remark \[rem:root\_isotopy\], we may assume that $h$ preserves 