Let S and S' be closed surfaces, then it is well known that for S' to cover S it is necessary and sufficient that chi(S)|chi(S'). (Here 'chi' denotes the Euler characteristic).

However, if S and S' are punctured surfaces then the above condition is necessary but no longer sufficient.

Is the question of determining necessary and sufficient criteria for S' to cover S answered in the research or expository literature?

I think that I know such criteria and want to use them a paper that I'm working on but (a) may be deceiving myself and (b) want to know whether I should write up a proof anew or whether there's a suitable reference. Surely the relevant criteria have been rediscovered many times, but I've never seen them discussed in writing.