Let $(X,\mu)$ be an $H$-space. The multiplication $\mu$ on $X$ allows us to define Pontrjagin product $$\cdot _\mu:H_k(X;R)\otimes H_l(X;R)\overset{\times}{\longrightarrow} H_{k+l}(X\times X;R)\overset{\mu_\ast}{\longrightarrow}H_{k+l}(X;R)$$ as the composition of homology cross product and the homomorphism induced by $\mu$. It seems that the definition of this product depends on the multiplication $\mu$, i.e., if $\mu'$ is another multiplication that makes $X$ an $H$-space, then potentially we have a different product operation. This leads to my question.
Question:
Let $\mu,\mu':X\times X\to X$ be multiplications that make $X$ an associative $H$-space, i.e., the multiplications are homotopy-associative, then $\mu$ and $\mu'$ gives $H_\ast(X;R)$ two ring structures. Is there a way to investigate the relationship between them? For instance, are they always isomorphic in some canonical way? Is there an example such that we have two non-isomorphic Pontrjagin rings associated with the same topological space equipped with different $H$-space structures?
Some trivial efforts:
If $X$ is an associative $H$-space with inverse, then by looking at the long exact sequence associated with the cofibration sequence $X\vee X\overset{i}{\to}X\times X\overset{q}{\to} X\wedge X$, we see that $\{\text{multiplications on $X$}\}$, which can be identified with a coset of $[X\times X,X]$, is in bijective correspondence with the group $[X\wedge X,X]$ of homotopy classes of (based) maps, where the group structure is given by the known multiplication $\mu$ on $X$. And we know that if $\mu'$ is another multiplication, then there exists $f\in [X\wedge X,X]$ such that $\mu'=\mu(\mu\times qf)\Delta_{X\times X}$, where $\Delta_{X\times X}:X\times X\to (X\times X)^2$ is the diagonal map.
I have only been able to understand elementary examples such as $S^1$ and $S^3$. For $X=S^1$, there is essentially one homotopy class of multiplicative structures inherited from $\Bbb C$, which is not interesting. For $S^3$, according to this paper by I. M. James, there are $8$ homotopy classes of associative multiplications, but they all induce the same ring structure on $H_\ast(X;R)$ for degree reasons.
There are plenty of associative $H$-spaces with inverse (e.g. loop spaces), but very few can be handled properly.
Consider two multiplications on $SO(3)\approx \Bbb RP^3$ given by $\mu_1(A,B)=AB$ and $\mu_2(A,B)=BA$. We note that $\mu_1\not\simeq\mu_2$ because they can be lifted to multiplications on $S^3$ as quaternion multiplications which are known to be not homotopic by the linked paper. We also note that $\mu_2=\mu_1\tau$, where $\tau(A,B)=(B,A)$ (i.e., $\tau$ swaps the two factors). With some help of homology cross product (and Künneth theorem), we see that $(-1)^{|\alpha||\beta|}\alpha\cdot_{\mu_1}\beta=(\alpha\cdot_{\mu_2}\beta)$ for $\alpha,\beta\in H_\ast(SO(3);R)$, so the multiplicative structures induced by $\mu_1,\mu_2$ are isomorphic.