On a functional equation related to projections of abelian groups

Summary.

For an abelian group (G, +, 0) we consider the functional equation

$$f:G \to G,\;f(x + y + f(y)) = f(x) + 2f(y)\quad (\forall x,y \in G),$$

((1))

most times together with the condition f(0) = 0. A solution of (1) is always idempotent. Our main question is as to whether it must be additive, i.e., a projection of the abelian group G.