Let $\Gamma \subseteq PSL_2(\mathbb{R})$ be a Fuchsian group, possibly containing elliptic elements. Is it true that $N(\Gamma) / \Gamma$, where $N(\Gamma)$ the normalizer of $\Gamma$ in $PSL_2(\mathbb{R})$, is isomorphic to the automorphism group of $\Gamma \backslash \mathcal{H}^*$?

Here, $\mathcal{H}^*$ is the union of the upper-half plane and the set of cusps of $\Gamma$.

If so, can you point me to a reference, please?