Given a finitely generated abelian group $\Gamma$, there are several way of realizing it as the $\tilde{K}_0$ of a $C^*-$algebra $A$.
One way to do this is by patching up Cuntz algebras for the torsion parts and say, compact operators for the free part. One can also use drop algebras (Section 13.1 of Rordam-Larsen-Lautsen).
My question is the following: can we find a locally compact group $G$ such that $\tilde{K}_0(C^*_{r}(G))\cong \Gamma$ ?
PS: I would be happy by a partial answer, for instance the special case $\Gamma=\mathbb{Z}/n\mathbb{Z}$.