An alternative functional equation related to the quadratic equation

Given a rational number = 2, we establish a criterion for the existence of the general solution of an alternative quadratic functional equation of the formwhere f is a mapping from an abelian group (G, ) to a uniquely divisible abelian group (H, +).We also find the general solution in the cases when G is a 6-divisible abelian group and G is a cyclic group.