We show the Teichm\"uller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichm\"uller space of its orientable double cover. Also, it is well known that the mapping class group $\text{ Mod} (N_g; k)$ of a non-orientable surface can be identified with a subgroup of $\text{ Mod} (S_{ g-1}; 2k)$, the mapping class group of its orientable double cover. These facts together with the classical Nielsen realization theorem are used to prove that every finite subgroup of $\text{ Mod}(N_g; k)$ can be lifted isomorphically to a subgroup of the group of diffeomorphisms $\text{ Diff}(N_g; k)$. In contrast, we show the projection $\text{ Diff}(N_g) \to \text{ Mod}(N_g)$ does not admit a section for large $g$.
Categories: math.AT