Nonamenable groups whose reduced group C*-algebras are not pure

We exhibit nonamenable groups whose reduced group C*-algebras are not pure. More precisely, if $\mathrm{\Gamma }$ is any countably infinite discrete group, then the reduced group C*-algebra of the restricted wreath product $(\mathbb{Z}\mathrm{/}2\mathbb{Z})\wr \mathrm{\Gamma }$ has an ideal-quotient isomorphic to $\mathcal{K}({\mathrm{\ell }}^2(\mathrm{\Gamma }))$. It is therefore not nowhere scattered and, in particular, not pure. Taking $\mathrm{\Gamma }$ nonamenable gives a negative answer to a question of Thiel concerning pureness of reduced group C*-algebras.