TITLE: Two H-space structures on S^3 and [X,S^3] different as groups for each: Explicit Example?
QUESTION [10 upvotes]: 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.

REPLY [5 votes]: This is semiexplicit.  For any H-space  $G$ with multiplication $\mu$, 
the projection maps $p_1, p_2: G\times G\to G$
have the property that 
$$
[p_1] \cdot [p_2] = [\mu] \in [G\times G, G].
$$
So if you have two different multiplications $\mu_1$ and $\mu_2$ on $S^3$ with induced
multiplications $*_1$ and $*_2$ on $[-,S^3]$, you'll have
$$
[p_1]*_1 [p_2] = [\mu_1] \neq  [\mu_2] = [p_1]*_2 [p_2]
$$
in $[S^3\times S^3, S^3]$.