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