Group amalgams of index (4, 2) and 4-valent 2-arc-transitive graphs

A finite group amalgam of index (k,m) is a 5-tuple (L,φ,B, ψ,R) where L, B, and R are finite groups, φ : B → L and ψ : B → R are monomorphisms, and the indices of B and B in L and R are k and m, respectively. The amalgam is faithful if N ≤ B, N C L, and N C R implies N = 1. It is well known that the finite faithful amalgams of index (k, 2) are in a bijective correspondence with (the conjugacy classes of) those arctransitive groups of automorphisms of the infinite k-valent tree whose vertex-stabilisers are finite. In this relationship, the 2-arc-transitive groups of automorphisms correspond to the amalgams (L,φ,B, ψ,R) for which the action of L on the cosets of B is 2-transitive. We call such amalgams 2-transitive. In this paper, we determine all finite faithful 2-transitive amalgams of index (4, 2). As a byproduct, we obtain a complete list of all 2-arc-transitive 4-valent graphs on no more than 512 vertices.