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.



Sufficient condition for coverings between non-orientable surfaces
Your necessary condition, rephrased slightly, is sufficient. It should say that the Euler characteristic of $X\_{k'}$ is a positive multiple of $\chi(X\_k)$. This modification is to take care of the possibility that $X\_k$ is $RP^2$, which is not covered by any other non-orientable surface. If $\chi(X\_k)= \chi(X\_{k'}) = 0$, then both are Klein bottles, and certainly there is a covering. Finally if $\chi(X\_k)<0$, then it is a connected sum of a torus with some number of projective planes. So it has non-orientable covers of all degrees; these are determined by their Euler characteristics. So, given $X\_{k'}$ with $\chi(X\_{k'}) = n \chi(X\_k)<0$, choose a degree n cover of $X\_{k}$; it will be homeomorphic to $X\_{k'}$. 


The case of orientable surfaces follows by the same argument.


