I will use the notation of this question. So, if $X$ is a (nice) topological space and $G$ is an abelian group, we can form its $G$-linearization $G[X]$. In McCord's article, this was denoted $B(G,X)$.

In that question it is mentioned that $\pi_*(G[X])=\tilde{H}_*(X;G)$. But more is true: the functor from Spaces to Graded Abelian Groups that maps a space $X$ to $\pi_*(G[X])$ is a homology theory, isomorphic to singular homology with coefficients in $G$. This says that the isomorphism commutes with the boundary maps in long exact sequences. And we have a good grip on what the boundary for $\pi_*(G[-])$ is, actually: if $A\to X$ is a cofibration, then McCord proves that

$$G[A]\to G[X]\to G[X/A]$$

is a fibration (actually, a principal bundle). The boundary map in homology for $A\to X$ corresponds to the boundary map in homotopy for this fibration.

With this set up, we have that $H_*(-;G)$ is naturally isomorphic to the composition $\pi_* \circ G[-]$.

If I understand correctly what I've heard, it is actually true that for any spectrum $E$ we have that $E_*$ (the associated homology functor from Spaces to Graded Abelian Groups) decomposes as a composition $\pi_* \circ E[-]$, where $E[-]$ is some functor from spaces to spaces which maps cofibrations to fibrations.

Question 1: what further hypotheses do we need on these functors to make the correspondence with the category of homology theories be a 1-1 correspondence? We certainly need $E[-]$ to map a point to a point (or to something contractible). Is this enough?

So, for $E=HG$, the Eilenberg-Mac Lane spectrum of $G$, we have that $E[-]=G[-]$: for $H\mathbb Z$, for example, $E$ is the infinite symmetric product functor.

For $E$ the sphere spectrum, we have $E[-]=Q$.

Question 2: what can be said about other spectra? Is there a clean description on how to get $E[-]$ from $E$ in general? Or maybe in other well-known cases, $KO, KU, MO, MU, K(n)$, etc...

Subsidiary question for the comments: any further references to this point of view are welcome.

definedto be $E_*(X) = \pi_* (E \wedge \Sigma^\infty X_+)$, where $X_+$ is $X$ with a disjointly added marked point, $\Sigma^\infty$ is stabilization and $\wedge$ is the smash product of spectra. Note that homology theories are ill-behaved in general, we can\t guarantee their representability in above form and even if representable the representation is generally highly non-unique.cont...$\endgroup$ – Anton Fetisov Jan 9 '16 at 13:50reduced 1-excisive functors(Goodwillie calculusis another good keyword). Note that this is going to give you the notion ofspectrum, which is much better behaved than the notion of (co)homology theory. $\endgroup$ – Denis Nardin Jan 9 '16 at 17:13