Latin squares with highly transitive automorphism groups

A Latin square is considered to be a set of n^2 cells with three block systems. An automorphism is a permutation of the cells which preserves each block system. The automorphism group of a Latin square necessarily has at least 4 orbits on unordered pairs of cells if n > 2. It is shown that there are exactly 4 orbits if and only if the square is the composition table of an elementary abelian 2-group or the cyclic group of order 3.