I have a 2-dimensional orbifold that is the quotient of $\mathbb{R}^2$ under a group of isometries $\Gamma$ generated by $180^\circ$ rotations. I would like to say that $\text{Isom}(\Gamma \backslash \mathbb{R}^2)$ is the same as $N(\Gamma)/\Gamma$, where $N(\Gamma)$ is the normalizer of $\Gamma$ in $\text{Isom}(\mathbb{R}^2)$. Proposition 8.6 in Metric Spaces of Non-Positive Curvature by Martin Bridson and Andre Hafliger says almost exactly this, but it is stated for groups acting freely. I think the proof should carry over but I'd love to get a reference for this and related facts.
