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. Let Γ be a finite graph and let Aut(Γ) be the automorphism group of Γ. It is well known that we can always find a finite graph Γ and a regular covering projection ℘: Γ∼ → Γ such that Aut(Γ) lifts along ℘. However, for constructing peculiar examples and in applications it is often important, given a subgroup G of Aut(Γ), to find ℘ along which G lifts but no further automorphism of Γ does, or even that Aut(Γ) ∼ is the lift of G. In this paper, we address these problems