  Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We
study the graph $\Gamma(\mathcal{C})$ 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
$\Gamma(\mathcal{C})$ 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.
