Let $f:X\rightarrow Y$ be a map of infinite loop spaces such that image of homology groups $H_i(X,\mathbb{Z})$ for $i\geq 1$ under $f_*$ are finitely generated. Does this imply that the image of homotopy groups under $f_*$ are also finitely generated? Few tips that might be helpful:
This is true for $\pi_1$ since it is Abelian.
This is true rationally.
In my case $X\rightarrow Y$ happens to be a covering space map too. So feel free to use that fact.
There is a counter-example if I replace the infinite loop space by $H$-space and you can see it here.