What do you mean by a covering?  

It sounds like you're talking about the action of the fundamental group of the base space on $\pi_0$ of the fibre, for a covering space.  Or some variant of it for a branched covering space.  

Take a look in Hatcher's Algebraic Topology textbook in Section 1.3, "Representating Covering Spaces By Permutations". 

Typically a branched covering space means a map $f : A \to B$ such that there is a co-dimension two subspace $A' \subset A$ such that $f$ when restricted to $A \setminus A' \to B \setminus f(A')$ is a covering space (locally trivial fibre bundle with discrete fibre), and $f$ in a regular neighbourhood of $A' \subset A$ usually satisfies additional constraints. Some authors can be pretty flexible on this.  Usually you want a tubular neighbourhood, and the mapping on the level of normal bundles is modelled on the map $S^1 \ni z \longmapsto z^n \in S^1$.