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 $A_5,A_6,PSL(2,7)$ or $PSL(2,8)$. We also consider the case when $\omega(G) = 6$andshowthatif$G$isanonsolvablefinitegroupwith$\omega(G) = 6$, then either $G\simeq PSL(3,4)$ or there exists a characteristic elementary abelian $2$-subgroup $N$ of $G$ such that $G\mathrm{/}N\simeq A_5$.