An extension of the sine addition formula on groups and semigroups

The functional equation $f(xy)=f(x)g(y)+g(x)f(y)$ is called the sine addition formula, and in a very general setting it is known that $g$ must be the average of two multiplicative functions. Here we consider the case in which the two multiplicative functions coincide, but we generalize that case to a functional equation with four unknown functions. That is, assuming that $M$ is a nonzero multiplicative function, we solve $f(xy)=k(x)M(y)+g(x)h(y)$ for the four unknown functions $f,g,h,k$ on groups and certain semigroups under the additional assumption that the unknown functions are at least central.