Let C be a conjugacy class of involutions in a group G. We study the graph
Γ(C) whose vertices are elements of C with g, h ∈ C connected by an edge if and only if
gh ∈ C. For t ∈ C, we define the component group of t to be the subgroup of G generated
by all vertices in Γ(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.