Variations on the Drygas equation and its stability

We study the stability of the Drygas functional equation: g(xy)+g(xy−1)=2g(x)+g(y)+g(y−1)g(xy)+g(xy−1)=2g(x)+g(y)+g(y−1) mixing the direct method of the proof with the method of the invariant means.Dropping the assumption about the domain to be an Abelian group, we assume that the function we are dealing with is central (i.e., g(xy)=g(yx)g(xy)=g(yx)), is approximatively central (i.e., |g(xy)−g(yx)|≤δ|g(xy)−g(yx)|≤δ), satisfies the Kannappan condition (i.e., g(xyz)=g(xzy)g(xyz)=g(xzy)), or that the group is amenable.