The Nielsen realization problem for non-orientable surfaces

We show the Teichmüller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichmüller space of its orientable double cover. It is also well known that the mapping class group $\mathrm{Mod}(N_g;k)$ of a non-orientable surface can be identified with a subgroup of $\mathrm{Mod}(S_{g-1};2k)$, 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 $\mathrm{Mod}(N_g;k)$ can be lifted isomorphically to a subgroup of the group of diffeomorphisms $\mathrm{Diff}(N_g;k)$. In contrast, we show the projection $\mathrm{Diff}\left(N_g\right)\to \mathrm{Mod}\left(N_g\right)$ does not admit a section for large g.