Non-split linear sharply 2-transitive groups

We give examples of countable linear groups $\Gamma < \operatorname {SL}_{{\mathbf {3}}}({\mathbf {R}})$, with no nontrivial normal abelian subgroups, that admit a faithful sharply $2$-transitive action on set. Without the linearity assumption, such were recently constructed by Rips, Segev, and Tent in [J. Eur. Math. Soc. 19 (2017), pp. 2895â??2910]. Our are permutational characteristic $2$, sense involutions do not fix point action.