An arithmetic analog of Klein's classification of finite subgroups of ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$

Let $K$ be a number field with ring of integers ${\mathcal{O}}_K$. We describe and classify finite, flat, and linearly reductive subgroup schemes of ${\mathrm{S}\mathrm{L}}_2$ over $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}{\mathcal{O}}_K$. We also establish finiteness results for these group schemes, as well as density results for the associated quotient singularities.