Lifting subgroups of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2$ to ${\mathrm{S}\mathrm{L}}_2$ over local fields

Let $K$ be a non-archimedean local field. We show that discrete subgroups without 2-torsion in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(K)$ can always be lifted to ${\mathrm{S}\mathrm{L}}_2(K)$, and provide examples (when $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(K)\mathrm{\ne }2$) which cannot be lifted if either of these conditions is removed. We also briefly discuss lifting representations of groups into ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(K)$ to ${\mathrm{S}\mathrm{L}}_2(K)$.