Let $\Gamma \leq PSL_2(\mathbb{R})$ be a Fuchsian group. What is the relation between $N(\Gamma) = \{ \alpha \in PSL_2(\mathbb{R}) \mid \alpha \Gamma \alpha^{-1} = \Gamma \}$ and $Aut(\Gamma \backslash \mathcal{H})$, the group of automorphisms of the Riemann surface $\Gamma \backslash \mathcal{H}$?

Here $\mathcal{H}$ is the upper half-plane.