Consider the (reduced) homology functor $H\_\*$ from the category of spectra to the category of graded Abelian groups. I wanted to know whether there is a "section" of this functor, i.e., a functor $F$ from graded Abelian groups to spectra so that $H\_\*\circ F=Id$.


(I have very little experience in Algebraic Topology, so I am guessing the answer is "obviously yes" or "obviously no", but I don't know which, and cannot find any reference. One can define the functor $F$ on objects to be a wedge sum of Moore spaces, but I do not know if this can be made functorial.)


Clarification: We may make impose additional restrictions if it suits our interest. In my situation, I am only interested in (possibly shifted) finite suspension spectra (i.e., suspension spectra of finite CW complexes), and finitely generated graded Abelian groups.



Section of the homology functor on spectra
I thought I will post the only reference I have found about non-existence of Moore space functor as an answer. It is due to Carlsson, "A counterexample to a conjecture of Steenrod" <https://link.springer.com/article/10.1007%2FBF01393939> and there is a nice discussion on nLab <https://ncatlab.org/nlab/show/Moore+space#nonfunctoriality_of_the_construction>


