  Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are
called "automorphism orbits" of $G$, and the number of automorphism orbits of
$G$ is denoted by $\omega(G)$. In this paper we prove that if $G$ is an
FC-group with finitely many automorphism orbits, then the derived subgroup $G'$
is finite and $G$ admits a decomposition $G = Tor(G) \times D$, where $Tor(G)$
is the torsion subgroup of $G$ and $D$ is a divisible characteristic subgroup
of $Z(G)$. We also show that if $G$ is an infinite FC-group with $\omega(G)
\leqslant 8$, then either $G$ is soluble or $G \cong A_5 \times H$, where $H$
is an infinite abelian group with $\omega(H)=2$. Moreover, we describe the
structure of the infinite non-soluble FC-groups with at most eleven
automorphism orbits.
