  We confirm a conjecture of Quillen in the case of the mod $2$ cohomology of
arithmetic groups ${\rm
SL}_2({\mathcal{O}}_{\mathbb{Q}(\sqrt{-m})}[\frac{1}{2}]\thinspace)$, where
${\mathcal{O}}_{\mathbb{Q}(\sqrt{-m}\thinspace)}$ is an imaginary quadratic
ring of integers. To make explicit the free module structure on the cohomology
ring conjectured by Quillen, we compute the mod $2$ cohomology of ${\rm
SL}_2({\mathbb{Z}}[\sqrt{-2}\thinspace][\frac{1}{2}])$ via the amalgamated
decomposition of the latter group.
