Homomorphisms of rank-$1$ quotient divisible groups

Torsion-free quotient divisible groups were introduced by R. Beaumont and R. Pierce in 1961. In 1998, A. A. Fomin and W. Wickless defined quotient divisible mixed groups and proved that categories of mixed quotient divisible groups and finite-rank torsion-free groups with quasihomomorphisms as morphisms are dual. In studying finite-rank torsion-free groups, rank-$1$ torsion-free groups play an important role, as many problems that are being solved in this class come to them. This paper is devoted to the study of the homomorphism groups of rank-$1$ quotient divisible groups. The main achieved result is a full description of the homomorphism group in the language of characteristics.