A variant of the quadratic functional equation on semigroups

Let S be a semigroup, let H be an abelian group which is uniquely 2-divisible, and let σ be an involutive automorphism of S. We express the solutions f : S → H of the following variant of the quadratic functional equation f(xy) + f(σ(y)x) = 2f(x) + 2f(y), x, y ∈ S, in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation.