 Sylow 2-subgroup of PGL2 over F4

The translations $x\mapsto x+b$ for $b\in {\mathbb{F}}_4$ form a Sylow $2$-subgroup of $PG{L}_2({\mathbb{F}}_4)$. Every nonidentity translation fixes infinity and exchanges the four finite points in two transpositions, so its action lies in $A_5$.