Lifting a prescribed group of automorphisms of graphs

In this paper we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Here we describe with an example the problems we address and refer to the introductory section for the correct statements of our results. Let $P$ be the Petersen graph, say, and let $\wp:\tilde{P}\to P$ be a regular covering projection. With the current covering machinery, it is straightforward to find $\wp$ with the property that every subgroup of $\Aut(P)$ lifts via $\wp$. However, for constructing peculiar examples and in applications, this is usually not enough. Sometimes it is important, given a subgroup $G$ of $\Aut(P)$, to find $\wp$ along which $G$ lifts but no further automorphism of $P$ does. For instance, in this concrete example, it is interesting to find a covering of the Petersen graph lifting the alternating group $A_5$ but not the whole symmetric group $S_5$. (Recall that $\Aut(P)\cong S_5$.) Some other time it is important, given a subgroup $G$ of $\Aut(P)$, to find $\wp$ with the property that $\Aut(\tilde{P})$ is the lift of $G$. Typically, it is desirable to find $\wp$ satisfying both conditions. In a very broad sense, this might remind wallpaper patterns on surfaces: the group of symmetries of the dodecahedron is $S_5$, and there is a nice colouring of the dodecahedron (found also by Escher) whose group of symmetries is just $A_5$. In this paper, we address this problem in full generality.