A non-complete graph Г is said to be (G, 2)-distance transitive if G is a subgroup of the automorphism group of Г that is transitive on the vertex set of Г, and for any vertex u of Г, the stabilizer Gu is transitive on the sets of vertices at distance 1 and 2 from u. In this talk, we investigate the family of (G, 2)-distance transitive graphs that are not (G, 2)-arc transitive, and classify such graphs of valency not greater than 5.