Classifying finite simple groups with respect to the number of orbits under the action of the automorphism group

Let omega(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying omega(G) less than or equal to n. Then we classify all finite simple groups G such that omega(G) less than or equal to 17. The latter result was obtained by computational means, using the computer algebra system GAP.