When is Pontrjagin ring isomorphic to Cohomology ring (under coefficient $\Bbb{Z}_p$ and $\Bbb{Z}$)?

Question

This question is motivated by the result of Poincare duality,

Definition: Suppose $X$ is an $H$-space (or in particular, assume $X$ is a topological group that can be represented as a manifold...). We know that the Pontrjagin product is given by the formula $$H_i(X)\otimes H_j(X)\overset{\times}{\longrightarrow}H_{i+j}(X\times X)\longrightarrow H_{i+j}(X)$$

Now, I'm interested in the following question:

• What are the necessary and sufficient condition for the following isomorphism? $$H_*(X;R)\cong H^*(X;R)$$ where $H_*(X)$ is the Pontrjagin ring and $H^*(X)$ is the cohomology ring, both with coefficients in a ring $R$. In particular, I'm interested in the case $R=\Bbb{Z}$.

Moreover, can we actually construct this isomorphism using cap product component-wise?

---

I took a look on a few examples with $\Bbb{Z}$ coefficient, but most of them don't satisfy the criterion:

1. $n$-dim torus $T^n=\prod_{i=1}^nS^1$. It is a topological group with addition of angles as the operation. We know that $H^*(T^n;\Bbb{Z})\cong\Lambda[a_1,\ldots,a_n]$, and $H_*(T^n;\Bbb{Z})\cong\Lambda[a_1,\ldots,a_n]$ by direct computation using the property of dual Hopf algebra. So I think this is one example of that isomorphism. In other words, what kind of conditions does $X$ needs to satisfy in order to make the two rings isomorphic?
2. $S^1\subset\Bbb{C}$ (resp. $S^3\subset\Bbb{H}$) has an $H$-space structure, and $H^*(S^1;\Bbb{Z})\cong\Lambda[x^1]\cong\Lambda[x_1]\cong H_*(S^1;\Bbb{Z})$, similarly for $S^3\subset\Bbb{H}$. Because the dual of $$\Delta(x^1)^2=(x^1\otimes 1+1\otimes x^1)^2=(x^1)^2\otimes 1+x^1\otimes x^1-x\otimes x+1\otimes (x^1)^2=0$$ gives the clue of the product operation in the dual Hopf algebra, which implies that $\Lambda_\Bbb{Z}[x^i]$ is self-dual, when $|x^i|$ odd.
3. This is just a guess... I think maybe $SU(n)$ satisfies this under $\Bbb{Z}$ coefficient, because I have a feeling that it's somehow related to the complex projective space, but I need $SU(n)$'s cell structure to rigorously compute $H^*$ and $H_*$.

Running out of examples...

Because Hatcher mentions "dual Hopf algebra" in his book, I think it may be simpler to search for such algebras and then realize them as spaces. Thus, I think we need Hopf algebras that are self-dual, in this case... However, to make this dualization useful, $H^j(X;\Bbb{Z})$ is required to be free for all $j$. This completely blocks my attempt because there are too many things to consider...

For other coefficients group like $\Bbb{Z}_p$, I have no idea what's going to happen.

Edit:

As @Tyrone has pointed out, there is an answer in $\Bbb{Q}$ for finite $H$-space.

Thanks in advance! :)