Cohomology of spectrum from cohomology of levels.

Question

Let $X$ be a spectrum and suppose that $H^i (X_n; \mathbb{Z})$ is known for every $n$, and every $i$.

Can one use this data to construct the cohomology of $X$, $$H^i(X,\mathbb{Z}) = (H\mathbb{Z})^*(X) = [\Sigma^{-i} X, H\mathbb{Z}] $$