Let $G$ be an abelian group (if this helps, let's say also finite). Let $BG_+$ be the classifying space together with a disjoint base point. What are the stable homotopy groups $$\pi^s_m (BG_+) := colim_n \pi_{n+m} (S^n \wedge BG_+)?$$ for example for $m=2$.

(The little I know is this: for $G=1$, we get the stable homotopy groups of spheres. For general $G$ and $m=0$, we get $\mathbf Z$, for $m=1$, we get $G/[G,G] \times \mathbf Z/2$.)

Thank you!