Let \(\Gamma\) be any finite connected graph that admits a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. Let v and w be adjacent vertices, A the stabilizer of v, and B the stabilizer of \(\{\) v,w\(\}\). The pair (A,B) is the symmetry type of \(\Gamma\), introduced by \textit{D. Z. Djoković} [in Conf. Szeged 1978, Colloq. Math. Janos Bolyai 25, 95-118 (1981; Zbl 0485.05031)]. It is easy to see that (1) \(| A: A\cap B| =k\), (2) \(| B: A\cap B| =2\), and (3) no nontrivial subgroup of \(A\cap B\) is normal in both A and B. Conversely, the author uses a simple argument based on \textit{J.-P. Serre}'s theory of group actions on trees [Trees (1980; Zbl 0485.05031)] to show that, for any pair (A,B) of finite groups satisfying (1), (2), and (3), there is a graph of arbitrarily large girth with symmetry type (A,B).