Generalized cohomology and stable homotopy groups of spectra

Question

Suppose two spectra $E$ and $F$ have same stable homotopy groups $\pi_k$ for $k\geq0$, equivalently, $E^{-k}(S)=F^{-k}(S)$. If we replace the sphere spectrum $S$ by another spectrum $X$, I wonder if the statement is still true: $$E^{-k}(X)=F^{-k}(X), \text{for } k\geq0.$$

Answer 1

Assume that $[S^k, E] = 0$ for all $k \geq 0$, and assume that $[S^{-1}, E] \neq 0$. That is, assume that $E^0(S^{-1}) \neq 0$. Let $F$ be the zero spectrum. Then $E$ and $F$ have the same homotopy groups in non-negative dimensions, but they disagree in degree $0$ cohomology for the spectrum $S^{-1}$.