A generalization of d'Alembert's functional equation on semigroups

Given a semigroup $S$ generated by its squares equipped with an involutive automorphism $\sigma$ and a multiplicative function $\mu :S\to \mathbb{C}$ such that $\mu (x\sigma (x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equation.