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Let $G$ be a group and $H$ an abelian group. Let $J^{*}(G,H)$ be the set of solutions $f:G\to H$ of the Jensen functional equation $f\left(xy\right)+f\left(x{y}^{-1}\right)=2f\left(x\right)$ satisfying the condition $f\left(xyz\right)-f\left(xzy\right)=f\left(yz\right)-f\left(zy\right)$ for all $x,y,z\in G$. Let $Q^{*}(G,H)$ be the set of solutions $f:G\to H$ of the quadratic equation $f\left(xy\right)+f\left(x{y}^{-1}\right)=2f\left(x\right)+2f\left(y\right)$ satisfying the Kannappan condition $f\left(xyz\right)=f\left(xzy\right)$ for all $x,y,z\in G$. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution $f:G\to H$ of the Whitehead equation is of the form $4f=2\varphi +2\psi$, where $2\varphi \in J^{*}(G,H)$ and $2\psi \in Q^{*}(G,H)$. Moreover, if $H$ has the additional property that $2h=0$ implies $h=0$ for all $h\in H$, then every...