[Edit: in response to prof. Sullivan's answer, let me try to expand the question a bit.]

Let $G$ be a discrete group which is perfect (trivial abelianization) and let $BG \to BG^+$ denote Quillen's plus construction, i.e., an acyclic map to a simply connected space. $BG$ comes with a preferred basedpoint and I'll use the image in $BG^+$ as basepoint there. Stable homotopy of $BG^+$ is defined as $\pi_k^s(BG^+) = \mathrm{colim}_n \pi_{n+k} (S^n BG^+)$ and comes with a "Hurewicz map" $\pi_k(BG^+) \to \pi_k^s(BG^+)$. One easily shows

**Lemma**: If $BG^+$ is an infinite loop space then this Hurewicz map $\pi_k(BG^+) \to \pi_k^s(BG^+)$ is split injective.

The idea of the proof is that if $BG^+ = \Omega^\infty X$ for a spectrum $X$, then the map $(\Omega^\infty X) \hookrightarrow \Omega^\infty S^\infty (\Omega^\infty X)$ adjoint to the identity of $S^\infty \Omega^\infty X$ is split by $\Omega^\infty \mathrm{ev}: \Omega^\infty (S^\infty \Omega^\infty X) \to \Omega^\infty X$.

Now the fact that $BG \to BG^+$ is acyclic implies that it induces an isomorphism $\pi_k^s(BG) \to \pi_k^s(BG^+)$, so $\pi_k(BG^+)$ is identified with a summand of $\pi_k^s(BG)$. On the other hand, we may use Pontryagin-Thom to identify $\pi^s_k(BG)$ with pairs $(M,f)$ where $M$ is a stably framed compact $k$-manifold and $f: M/\partial M \to BG$ is a pointed map, up to cobordism of such pairs.

**Question:** Is there a simple criterion for when an element of $\pi_k^s(BG)$ is in the image of $\pi_k(BG^+) \to \pi_k^s(BG^+) = \pi_k^s(BG)$?

Now "simple criterion" is of course not very precise, so let me give an example. It is easy to see that (for $k \geq 2$) it is sufficient to be representable by $(M,f)$ with $M/\partial M$ a homology sphere. Indeed, we can plus-construct the map $f: M/\partial M \to BG$ to get $S^k \simeq (M/\partial M)^+ \to BG^+$. Is this simple and sufficient criterion also necessary? If not, is there some other "nice" description of this summand $\pi_k(BG^+) \subset \pi_k^s(BG^+) = \pi_k^s(BG)$?