Covering spaces of surfaces 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 $\Sigma_h$?
 A: For an explicit example of a non-cyclic group $G$ acting freely on a surface $S$ take the union $X$ of the edges of the 3-cube $[-1,1]^3\subset\mathbb{R}^3$; set $S$ to be the boundary of a small neighborhood of $X$, and take the group generated by the rotations through $\pi$about the coordinate axes in $\mathbb{R}^3$ as $G$; note that $G$ is isomorphic to Klein's group $\mathbb{Z}/2\oplus\mathbb{Z}/2$, the "simplest" non-cyclic group there is.
A: The OP is asking for a classification of finite index normal subgroups of fundamental groups of closed surface. This is hairy, but algorithmic, see Gareth Jones' 1994 math Scand paper.
A: 
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 $\Sigma_h$? 

In brief, the answer to this part of the question is 'You can take $M\to\Sigma_g$ to be regular, but beyond that, no.' This can already be extracted from Misha's comments above, but let me try to summarise.
First, note that there is a regular covering $M\to\Sigma_g$ that factors through $\Sigma_h\to\Sigma_g$.  (Specifically, you can take $\pi_1M$ to be the intersection of all the conjugates of $\pi_1\Sigma_h$.)  So, as in the earlier part of your question, you can take $\Sigma_h\to\Sigma_g$ to be regular.
You can now rephrase your question in terms of normal quotients of $\pi_1\Sigma_g$, and it becomes

Given any finite quotient $q:\pi_1\Sigma_g\to Q$, is there some kind of 'standard' finite quotient $p:\pi_1\Sigma_g\to P$ such that $q$ factors through $p$?

In particular, $P$ surjects $Q$.  But any finite group can arise as $Q$ (see Misha's and algori's comments---the point is that $\pi_1\Sigma_g$ surjects a free group), so you are looking for a 'standard' family of finite groups that surjects every finite group.  But there's no 'natural' definition of such a family. 
Remark: Obviously, there are such families, such as $\{ Q\times\mathbb{Z}/2\}$ where $Q$ is an arbitrary finite group, but clearly this is not 'natural'.
