Let $G$ be a finite group. Atiyah proved that the $K$-cohomology of $BG$ vanishes in odd degrees and in even degrees is the completion of the representation ring of $G$ at the augmentation ideal.


What is the $K$-homology of $BG$? I mean the stable homotopy groups of the complex $K$-theory spectrum smash $BG$.



$K$-homology of $BG$
In "$K$-homology of universal spaces and local cohomology of the representation ring" Greenlees proves that 
$$
K\_i(BG\_+) \simeq H\_J^i(R(G))
$$
for $i=0,1$, where $J$ is the augmentation ideal and $H\_J^i(R(G))$ refers to [local cohomology.](https://en.wikipedia.org/wiki/Local_cohomology) In fact Greenlees proves something more general using equivariant homotopy theory (for which the above is a special case), and this more general object is also studied in Chapter 18 of Greenlees and May's "Generalized Tate Cohomology"


Equivalently, one can use Anderson's universal coefficient theorem for $K$-theory (see [here](http://faculty.tcu.edu/gfriedman/notes/Anderson-UCT.pdf)). Anderson shows that if $G$ is even a compact connected Lie group that $KU\_1(BG) = 0$ and $$KU\_0BG = \operatorname{lim}\_n\operatorname{Hom}(R(G)/J^n,\mathbb{Z}).$$


There is a discussion of the relationship between the two results in Remark 2.8 of "Topological $K$-(co)homology of classifying spaces of discrete groups", by Joachim and Lück (which also contains more results along these lines). 


