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A spectrum is a topologist's abelian group, so the result is similar: you can only learn the stabilization $\Sigma^\infty G$. If you need $G$ itself, then a better choice would be the unstable equivariant category of $G$-spaces, but even then I am unsure if you can reconstruct $G$ itself and not just its homotopy type. You can probably do it for compact groups, but I don't have a reference.
@DenisNardin , I don't think they are equivalent. I suppose def. 2 uses a limit of sets rather than a homotopy colimit of spaces (since you can't define it without considering Fredholm space like in def. 3). Thus 2 and 3 are related by the Milnor exact sequence.
Mike's comment is wrong. An unoriented bundle is the same as a line bundle and an oriented bundle, so $BO(2) = BSO(2) × BZ/2 = B^2 Z × BZ/2$. Thus such bundles are completely classified by $w_1 \in H^1(Z/2)$ and a class in $H^2(Z)$ which maps to $w_2$ under $Z \to Z/2$. I suppose we should call it an integral Stiefel-Whitney class $\hat w_2$. If we look at cohomology then for $B^2 Z = \Bbb C P^\infty$ we have $H^*(Z) = Z[[x]]$ with $\mathrm{deg} x = 2$, $x = \hat w_2$. In particular $x^2 = p_1$.
@SebastianGoette Indiscrete spaces are a mock of topology. Of course I consider only sober spaces, which cover all real examples. But if you wish so, you could always label each open subset with its elements, it would be an absolute invariant.
A good-behaved object is a cohomology theory, where Brown's representability gives us a spectrum $E$ s.t. $E^*(X) = \pi_{-*}Hom(\Sigma^\infty X_+, E)$. Ordinary cohomology ($E^0 (pt) = G$, $E^n(pt) = 0$ if $n\ne 0$) are represented by Eilenberg-Maclane spectra $HG$, while for all other spectra $E^*(pt)$ is very nontrivial.
I am not sure what are you asking. If $E$ is a spectrum, then the homology theory $E_*$ is defined to 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...
I'd say that it is a locally cartesian closed $(\infty,1)$-category, such that the groupoid of fibrations satisfies descent. There are two problems: what is its internal logic and what exactly is meant by descent (since with an arbitrary elementary topos it is difficult to talk about diagram size). The best version of the second condition that I can state is that the functor $X\mapsto E/X: E^{op} \to Top$ is absolutely flat (a substitute for representability). The first condition should be satisfied by some version of HoTT (not the current one, it is interpretable only in model categories).
I like your approach. Two notes: $Be \to K(Z, 4)$ should be not $\mathbb G_m$ but rather $B\mathbb G_m$ with a suitable rigid structure (over complex numbers it should be $\mathbb C(x)^* / \mathbb C^*$). Also I think maps $BG \to K(Z, 4)$ may be not rigid enough, we need to actually get a group extension by $B\mathbb G_m$.
