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Is Eilenberg-Maclane $\wedge$ Moore space the spectrum of the cohomology theory $H^*(\ ,G)$?
In case you are curious, here is a more concise but higher level argument.
The homotopy groups of the spectrum $H\mathbb Z \wedge M_k(G)$ agree with the homology groups of $M_k(G)$, which is just a $ …