In the last chapter of May's Concise course, we have an isomorphism of $\mathbb{Z}_2$-cohomology of spectra, induced by a map of spectra. And then May writes, it can be deduced that the induced map on integer cohomology is an isomorphism as well from the fact, that the homotopy groups of both spectra are $\mathbb{Z}_2$-vector spaces. But he doesn't explain this deduction nor gives he a hint.
So, why does this hold? Or maybe, is there a more explicit treatment of this calculation?