Suppose that, given an abelian group $G$, there is a multiplication map $\mu:K(G,n)\times K(G,n) \to K(G,n)$ defined such that the induced map on the homotopy group $\mu_*:\pi_n(K(G,n) \times K(G,n)) \to \pi_n(K(G,n))$ takes $(g_1,g_2)$ to $g_1 + g_2$, where $+$ is the operation on $G$.
Does it follow that this multiplication is homotopy-commutative; that is, if $t:K(G,n) \times K(G,n) \to K(G,n) \times K(G,n)$ switches coordinates, does it follow that $\mu t$ is homotopic to $\mu$? Since $G$ is commutative, it seems that $\mu$ should be, but I'm having a hard time coming up with the actual homotopy. I know that is NOT true in general that if two maps induce the same homomorphisms on homotopy groups, then they are homotopic.
One could also ask if the fact that the operation on $G$ is associative implies that $\mu$ is homotopy-associative.