Let G be a group and H an abelian group. Let J ∗ (G, H) be the set of solutions f : G → H of the Jensen functional equation f(xy) + f(xy−1 ) = 2f(x) satisfying the condition f(xyz) − f(xzy) = f(yz) − f(zy) for all x, y,z ∈ G. Let Q ∗ (G, H) be the set of solutions f : G → H of the quadratic equation f(xy) + f(xy−1 ) = 2f(x) + 2f(y) satisfying the Kannappan condition f(xyz) = f(xzy) for all x,y, z ∈ G. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution f : G → H of the Whitehead equation is of the form 4f = 2ϕ + 2ψ, where 2ϕ ∈ J ∗ (G, H) and 2ψ ∈ Q ∗ (G, H). Moreover, if H has the additional property that 2h = 0 implies h = 0 for all h ∈ H, then every solution f : G → H of the Whitehead equation is of the form 2f = ϕ+ψ, where ϕ ∈ J ∗ (G, H) and 2ψ(x) = B(x,x) for some symmetric bihomomorphism B : G × G → H.