 Unipotent involutions in characteristic two

An involution in characteristic two has Jordan blocks of sizes at most two. Its conjugacy type is determined by the rank of $N$, subject to $2\mathrm{rank}N\le \dim V$. GL-conjugacy can split in SL if the centralizer determinant map is not onto; that issue must be checked separately. In $S{L}_3(4)$ a one-dimensional untouched block makes that determinant map onto, yielding one involution class. In $S{L}_4(2)=G{L}_4(2)$ the two possible nonzero ranks give two classes. Their projective quotients have equal order but different involution class counts.