Infinite locally dihedral groups as automorphism groups

If A is a nontrivial torsion-free, locally cyclic group with no nontrivial divisible quotients, and G is the split extension of A by a group of order 2 acting on A by means of the inverting map, then G≃AutG . We prove that in no other case the full automorphism group of a group is infinite and locally dihedral.