Let $X_k$ be the connected sum of $k$ projective planes. I am interested in necessary and sufficient conditions for the existence of a covering $\pi: X_{k'} \to X_k$, where $k$ and $k'$ are integers.

A necessary condition is that the Euler characteristic of $X_{k'}$ is a multiple of the Euler characteristic of $X_k$. Though obtaining a sufficient condition seems more difficult.

The case of orientable surfaces is easy, but this is more difficult than I think, and I can't find it anywhere.

I would appreciate any hint about this.