The following is offered as main result. Let \((G,\cdot)\) and \((H,+)\) be abelian groups, and \(e\) the neutral element of \((G,\cdot)\). The solutions \(f: G\to H\) of \(f(xy)+f(xy^{-1})=2f(x)\), \(f(e)=0\) are exactly the homomorphisms of \(G\to H\) if, and only if, either \(H\) has no element of order 2 or \([G:G^ 2]\leq 2\), where \(G^ 2:=\{x^ 2\mid\;x\in G\}\). While this is not true in general for nonabelian groups, a partial result (in the ``only if'' direction) is presented in this case too.