Let $S_{g,b}$ denote the orientable connected compact surface of genus $g$ with $b$ boundary components. A group homomorphism $\varphi\colon G\to \text{Homeo}^+(S_{g,b})$ is said to be free $G$-action if $\varphi(a)$ has no fixed point for all non-trivial $a\in G$. Two free group actions $\varphi_1,\varphi_2\colon G\to \text{Homeo}^+(S_{g,b})$ are said to be equivalent if there is $\mathscr H\in \text{Homeo}^+(S_{g,b})$ such that $\varphi_2(a)=\mathscr H^{-1}\circ \varphi_1(a)\circ \mathscr H$ for all $a\in G$.

A theorem of Nielsen says that any two free $\Bbb Z/n\Bbb Z$-actions on a closed orientable connected surface are equivalent.

Does there exist a classification theory of inequivalent free $\Bbb Z/n\Bbb Z$-actions on every $S_{g,b}\ (b\neq 0)$?

Any reference/idea will be helpful.

Let $S_{g,b}$ denote the orientable connected compact surface of genus $g$ with $b$ boundary components. A group homomorphism $\varphi\colon G\to \text{Homeo}^+(S_{g,b})$ is said to be free $G$-action if $\varphi(a)$ has no fixed point for all non-trivial $a\in G$. Two free group actions $\varphi_1,\varphi_2\colon G\to \text{Homeo}^+(S_{g,b})$ are said to be equivalent if there is $\mathscr H\in \text{Homeo}^+(S_{g,b})$ such that $\varphi_2(a)=\mathscr H^{-1}\circ \varphi_1(a)\circ \mathscr H$ for all $a\in G$.

A theorem of Nielsen says that any two free $\Bbb Z/n\Bbb Z$-actions on a closed orientable connected surface are equivalent.

Does there exist a classification theory of inequivalent free $\Bbb Z/n\Bbb Z$-actions on every $S_{g,b}\ (b\neq 0)$?

Any reference/idea will be helpful.