Let $\Sigma_g$ be a surface of genus $g\ge 2$, and let $\Sigma_k$ be an $m$-sheeted covering space of $\Sigma_g$. It is known that $k=m(g-1)+1$.

An example of such a covering space is a regular covering obtained by choosing one ``hole" as the center of the symmetry and take $\Sigma_k$ to have $m$-fold rotational symmetry around that chosen center (as on standard pictures in your favorite topology book).

**Question:** Does every finite sheeted regular covering space of $\Sigma_g$ arise in this way?

It feels like this should be known/standard, but I can't find an argument or a reference.

**Edit**: I agree, the question is not very precise as stated, I certainly didn't have in mind that every finite cover arises via cyclic symmetry as in the example above. The reason for the lack of precision is that I do not have a particular result in mind, but I would like to know if there is a simple geometric description of the covering *map*, as Misha points out, between two surfaces? The example with the cyclic group gives such a simple description of the covering map.
Or, given any covering map $\Sigma_h\to \Sigma_g$ between two surfaces, is there some kind of a ``standard" covering $M\to \Sigma_g$, which factors through $\Sigma_h$?