It is well-known that for two (discrete) abelian groups $G$ and $H$, the set $[K(G,n),K(H,n)]_*$ of based homotopy classes of maps between the corresponding Eilenberg-MacLane spaces is in canonical bijection with ${\rm Hom}_{\rm Ab}(G,H)$, the set of group homomorphisms from $G$ to $H$. The usual textbook proof uses that $[X,K(H,n)]_* \cong H^n(X;H)$ and then uses the Universal Coefficients Theorem and Hurewicz to see $H^n(K(G,n);H) \cong {\rm Hom}_{\rm Ab}(G,H)$.

Question: Do we have a similar result if we replace homotopy classes of based maps $K(G,n) \to K(H,n)$ with homotopy classes ofgroup homomorphisms$K(G,n) \to K(H,n)$?

The model I want to take for $K(G,n)$ is the $G$-linearization $G[S^n]$ of $S^n$, introduced by McCord as $B(G,S^n)$. Its elements are formal sums $\sum_{i=1}^ng_i x_i$ with $g_i \in G$ and $x_i \in S^n$, with the expected identifications and topology. It is a topological abelian group by adding formal sums.

That this topological group is indeed a model for $K(G,n)$ follows from a theorem of McCord (1969), which gives us a fiber sequence $G[S^{n-1}] \to G[D^n] \to G[S^n]$ induced by the cofiber sequence $S^{n-1} \hookrightarrow D^n \to D^n/S^{n-1} \cong S^n$. We conclude that there is a weak equivalence $G[S^{n-1}] \to \Omega(G[S^n])$ and by induction $G \simeq \Omega^n(G[S^n])$, showing that $G[S^n]$ is a $K(G,n)$ as desired.

Now a more precise formulation of my question is:

Question: Do we have a bijection $\pi_0({\rm Hom}_{\rm TopAb}(G[S^n],H[S^n])) \cong {\rm Hom}_{\rm Ab}(G,H)$?

There are obvious candidates for the maps in both directions. From left to right, we send $f: G[S^n] \to H[S^n]$ to $\pi_n(f): G \to H$. Conversely, a group homomorphism $\phi: G \to H$ is sent to $\phi[S^n]: G[S^n] \to H[S^n]$. Clearly $\pi_n(\phi[S^n]) = \phi$ for all $\phi: G \to H$. However, it is not so clear whether every continuous group homomorphism $f: G[S^n] \to H[S^n]$ is homotopic (through group homomorphisms) to one of the form $\phi[S^n]$.

**Possible approach**: Since we have an adjunction ${\rm Hom}_{\rm TopAb}(G[S^n],H[S^n]) \cong {\rm Hom}_{\rm TopAb}(G,\Omega^n(H[S^n]))$, where $\Omega^n(H[S^n])$ has the pointwise abelian group structure, it would suffice to prove that ${\rm Hom}_{\rm TopAb}(G,-)$ preserves the weak equivalence $H \xrightarrow{\simeq} \Omega^n(H[S^n])$. My idea would be to do this in the following steps:

- This is clear for $G = \mathbb{Z}$ since ${\rm Hom}(\mathbb{Z},A) \cong A$.
- Similarly, for $G = \mathbb{Z}/k$, we can use that ${\rm Hom}(\mathbb{Z}/k,A) \cong {\rm Tor}_k(A)$ and apply the above result to the abelian group ${\rm Tor}_k(H)$ instead of $H$.
- This means the result is true for any finitely generated $G$.
- By writing $G$ as colimit over its f.g. subgroups, it follows for any $G$.

However, I have not been able to find this result in the literature and thus I'm fearing I'm overlooking something. Can someone tell me whether this approach works? Also a reference to a similar result in the literature would be nice.