x*y = f(a) + g(b) + c,where c is a fixed element and f,g are automorpisms of this Abelian group, such that f(g(x))=g(f(x)). Then (G,*) is a medial quasigroup. Here is a finite example:
* | a b c d e f g --+---------------------- a | a d g c f b e b | c f b e a d g c | e a d g c f b d | g c f b e a d e | b e a d g c f f | d g c f b e a g | f b e a d g c