Unexpected subgroup schemes of PGL_{2,k} in characteristic 2

If the characteristic of a field $k$ is odd any infinitesimal group scheme of $PG{L}_{2,k}$ lifts to $S{L}_{2,k}$. 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 $PG{L}_{2,k}$. Also, the infinitesimal trigonalizable case is considered.