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We prove that if G is a group such that Aut G is a countably infinite torsion FC-group, then Aut G contains an infinite locally soluble, normal subgroup and hence a nontrivial abelian normal subgroup. It follows that a countably infinite subdirect product of nontrivial finite groups, of which only finitely many have nontrivial abelian normal subgroups, is not the automorphism group of any group.doi:10.1090/s0002-9939-1986-0813805-2 fatcat:ecz7e7snjneg5ceyk446henige