A question about the Euler characteristic

Question

Let $G$ be a finite group acting freely on a compact and orientable Riemannian manifold of dimension 2. I want to show that $\chi(M/ G)=\frac{\chi(M)}{|G|}$, where $\chi$ is the Euler characteristic, just using techniques in the scope of "elementary" Riemannian Geometry. I thought about Gauss-Bonnet Theorem, but I'm not able to use it properly. Can you help me ? Thank you.