We investigate the computational complexity of the problem of deciding if an algebra homomorphism can be factored through an intermediate algebra. Specifically, we fix an algebraic language, , and take as input an algebra homomorphism between two finite -algebras and , along with an intermediate finite -algebra . The decision problem asks whether there are homomorphisms and such that . We show that this problem is NP-complete for most languages. We also investigate special cases where homomorphism factorization can be performed in polynomial time.