ON FINITE GROUPS WITH FEW AUTOMORPHISM ORBITS

Denote by omega(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which omega(G) <= 5, then G is isomorphic to one of the groups A(5), A6, PSL(2, 7), or PSL(2, 8). We also consider the case when omega(G) = 6 and show that, if G is a nonsolvable finite group with omega(G) = 6, then either G similar or equal to PSL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N similar or equal to A(5).