There are *twelve* continuous maps $S^3\times S^3\to S^3$ up to homotopy that make the three-sphere $S^3$ into an H-space. This follows from a result of James [1], which says that if there exists one such multiplication on $S^n$ then the homotopy classes of multiplications that make $S^n$ into an H-space are in bijection with $\pi_{2n}(S^n)$. For $n=3$, we have $\pi_6(S^3) \cong \mathbb{Z}/12$. Not all of these other multiplications are homotopy associative, but eight of them are, and hence necessarily also have homotopy inverses.

Question.Is there an example in the literature of a space $X$, and two homotopy associative multiplications $m,m':S^3\times S^3\to S^3$, such that the groups $[X,(S^3,m)]$ and $[X,(S^3,m')]$ have been calculated explicitly and are not isomorphic? Necessarily, $X$ cannot be a suspension.

I am also interested in the same question but with $S^3$ replaced with any other H-space.

[1] James, I. M. "Multiplication on spheres (II)." Transactions of the American Mathematical Society (1957): 545-558.