A theorem of W. Browder [1] from 1961 says that an H-space $X$ with finitely generated integral homology has $\pi_2(X) = 0$. This generalizes Cartan's famous theorem that $\pi_2$ of a Lie group vanishes.

I would like to know if there is a relative version of the following sort: Suppose that $f: X \to Y$ is an H-space map and $\ker(H_*(X) \to H_*(Y))$ is finitely generated. Then can I conclude that $\ker(\pi_2(X) \to \pi_2(Y))$ is $0$?

[1] Torsion in H-Spaces, Annals of Mathematics, Second Series, Vol. 74, No. 1 (Jul., 1961), pp. 24-51. http://www.jstor.org/stable/1970305 .