Let me begin by a couple of questions :

Consider a graded abelian group $V=\oplus_{i\geq 0} V_i$ such that $V_{2i}=\mathbb{Z}$ and $V_{\textrm{odd}}=0$. What are the possible Hopf algebra structures on it?

One can ask a slightly stronger question :

When does a given Hopf algebra structure on $V$ (as above) arise as the integral cohomology of a $H$-space?

The motivation behind this question is purely my own curiosity. While discussing how to distinguish $\Omega S^3$ and $\mathbb{CP}^\infty$ rationally, we saw that the rational cohomology or the rational homotopy groups are unable to detect the difference. However, $H^\ast(\Omega S^3;\mathbb{Z})=\Gamma_{\mathbb{Z}}[\alpha]$, the divided polynomial algebra generated by $\alpha$ (of degree $2$) while $H^\ast(\mathbb{CP}^\infty;\mathbb{Z})=\mathbb{Z}[u]$ is the polynomial algebra generated by $u$ (of degree $2$). Moreover, a polynomial algebra such as $\mathbb{Z}[u]$ has a comultiplication map given by $u\stackrel{\Delta}{\longrightarrow}1\otimes u+u\otimes 1$ and extended naturally. One can check that the dual (as a Hopf algebra) of $\mathbb{Z}[u]$ is isomorphic to $\Gamma_{\mathbb{Z}}[u^\ast]$, where $u^\ast$ is the dual of $u$.

From what I could conclude by playing around with coefficients is that for each prime $p$ and a positive integer $r$ one can cook up a Hopf algebra structure on $V$. I don't know if they come from a space. However, these structure constants must be compatible with the action of the Steenrod algebra (or the mod $p$ version) if $V=H^\ast(X;\mathbb{Z})$ for some $H$-space $X$. I vaguely remember that compatibility with the Steenrod algebra is not sufficient and Adams operations provide further obstructions (although I may be wrong on this point). This leads me to :

Is there a (list of) necessary and sufficient criteria (in general or at least in this case) which tells us when a given Hopf algebra structure on graded vector space arises as $H^\ast(X;\mathbb{Z})$ for some $H$-space?

This may be well known (and perhaps classical) to homotopy theorists and any reference to known results are good enough for me.