Suppose we have two (affine) algebraic $\mathbb{Q}$-groups $G$ and $H$ and a $\mathbb{Q}$-morphism $\phi:G\rightarrow H$. Given a $\mathbb{Q}$-form ${}_{\xi}G$ of $G(\overline{\mathbb{Q}})$ is it true that $\phi(\overline{\mathbb{Q}})({}_{\xi}G(\mathbb{\mathbb{Q}}))$ lies in a $\mathbb{Q}$-form of $H(\overline{\mathbb{Q}})$?
For example, suppose $G$ is $SL_2\times SL_2$, $H$ is $SL_3\times SL_3$ and $\phi$ is $\tau_3\times \tau_3$ where $\tau_3:SL_2\rightarrow SL_3$ is the irreducible representation. Is it true that any $\mathbb{Q}$-form of $SL_2\times SL_2$ lies in a $\mathbb{Q}$-form of $SL_3\times SL_3$? I know this is true for inner forms as can be shown using the first Galois cohomology group description of $\mathbb{Q}$-forms.