Wilson's functional equation with an anti-endomorphism

Let M be a topological monoid, let psi : M -> M be a continuous anti-homomorphism of M, and let mu : M -> C be a continuous multiplicative function such that mu(x psi(x)) = 1 for all x is an element of M. We describe, in terms of multiplicative functions, additive functions and characters of 2-dimensional representations of M, the solutions (w, g), where w, g : M -> C, such that w is central and g is continuous, of the new functional equation w(xy) + mu(y)w(x psi(y)) = 2w(x)g(y), x, y is an element of M. We also treat the equation on compact groups and the special equation (when mu = g = 1): Jensen's functional equation.