The binary actions of simple groups of Lie type of characteristic 2

Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We study the graph $\mathrm{\Gamma }(\mathcal{C})$ whose vertices are elements of $\mathcal{C}$ with $g,h\in \mathcal{C}$ connected by an edge if and only if $gh\in \mathcal{C}$. For $t\in \mathcal{C}$, we define the component group of $t$ to be the subgroup of $G$ generated by all vertices in $\mathrm{\Gamma }(\mathcal{C})$ that lie in the connected component of the graph that contains $t$. We classify the component groups of all involutions in simple groups of Lie type over a field of characteristic $2$. We use this classification to partially classify the transitive binary actions of the simple groups of Lie type over a field of characteristic $2$ for which a point stabilizer has even order. The classification is complete unless the simple group in question is a symplectic or unitary group.