The authors study the solutions of the Whitehead functional equation \[ f(xyz) + f(x) + f(y) + f(z) = f(xy) + f(xz) + f(yz), \] where the function \(f\) is defined on a group and takes its values in an abelian group. The solutions of the Fréchet functional equation \[ f(xyz) + f(x) + f(y) + f(z) = f(xy) + f(zx) + f(yz) \] under the same conditions are also investigated. For the description of the solutions the authors use standard methods, reducing the equations to Jensen and quadratic equations.