  An open question is whether the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow
\widetilde{K_0 }\mathbb{Q} G$ in reduced $K$-theory from the integral to the
rational group ring is trivial for any group $G$. We will show that this is
false, with a counterexample given by the group $QD_{32} *_{Q_{16}} QD_{32}$.
We will also show how to compute the image of the map $\widetilde{K_0
}\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ using representation
theoretic means, assuming $G$ satisfies the Farrell-Jones conjecture.
