Let $f : (\mathbb{Q},+) \longrightarrow (\mathbb{Q},+)$ be a non-zero homomorphism.
Can we conclude that $f$ is bijective (or, if that fails, that $f$ is injective or surjective)?
Context
The additive group of integers has non-surjective nonzero endomorphisms, such as $n\mapsto 2n$. However, the same formula gives a bijective endomorphism when applied to rationals.