Unexpected subgroup schemes of PGL2,k in characteristic 2

If the characteristic of a field k is odd any infinitesimal group scheme of PGL2,k lifts to SL2,k. In this paper, we prove that this is not true in characteristic $2$ and we give a complete description, up to isomorphism, of infinitesimal unipotent subgroup schemes of PGL2,k. Also, the infinitesimal trigonalizable case is considered.