Given a regular covering projection between two connected graphs, one of the central question is what is the relationship between the automorphism groups of these two graphs. It is well-known that the covering graph may possess automorphisms that do not project to automorphisms of the base graph and also that the base graph may have automorphisms that do not lift to automorphisms of the covering graph. In this talk the following problem will be addressed: Given a connected finite graph X and a group of automorphisms G acting upon X, does there exist a regular covering projection to X such that the maximal group that lifts is G. I will present a recent result by Pablo Spiga and myself that shows that under some mild conditions on G and X the answer to this question is always affirmative. I will then also discuss the question whether the covering projection can be chosen in such a way that the full automorphism group of the covering graph equals the lift of the group G. |