Suppose that $G$ is an affine algebraic group defined over $\mathbb{Q}$. Then we can take its group $G(\mathbb{Q})$ of $\mathbb{Q}$-points, and abelianisation followed by rationalisation provides a homomorphism
$$\varphi\colon G(\mathbb{Q}) \longrightarrow G(\mathbb{Q})^\mathrm{ab} \otimes_{\mathbb{Z}} \mathbb{Q}$$
Since the right side is isomorphic to a direct sum of copies of $\mathbb{Q}$, it is also isomorphic to the $\mathbb{Q}$-points of an affine algebraic group. My question is whether this isomorphism can be chosen such that $\varphi$ is algebraic:
Is there a morphism $f \colon G \to H$ of affine algebraic groups defined over $\mathbb{Q}$ and an isomorphism $\lambda \colon H(\mathbb{Q}) \to G(\mathbb{Q})^\mathrm{ab} \otimes_{\mathbb{Z}} \mathbb{Q}$ such that $\lambda \circ f_\mathbb{Q} = \varphi$?
I naively expect the answer to be "no", and if this is the case a counterexample would be helpful.
