Let $C$ be a hyperelliptic curve of genus 2 defined over $\mathbb{Q}$, let $J$ be its Jacobian, and let $X$ be the Kummer surface associated to $J$ (i. e. $X$ is the singular Kummer surface which results from quotienting $J$ by the map $x \mapsto -x$, not the K3 surface which resolves the singularities of $X$). I have a few questions suggested by computations I have done in a specific instance of this setup. In my situation $X$ is defined over $\mathbb{Q}$ as a quadric hypersurface in $\mathbb{P}^3$. I don't know if it's relevant, but the Jacobian I am working with happens to have Mordell-Weil group $(\mathbb{Z} / 2 \mathbb{Z})^3$.

Let $\pi: J(\overline{\mathbb{Q}}) \to X(\overline{\mathbb{Q}})$ be the projection map. Computational evidence suggests that if $\pi(x) \in X(\mathbb{Q})$, then $\pi(n \cdot x) \in \mathbb{Q}$ for all $n \in \mathbb{Z}$. Is this known to be true?

More broadly, if $\pi(x), \pi(y) \in X(\mathbb{Q})$, is $\pi(x + y) \in X(\mathbb{Q})$? If so, is $\pi^{-1}(X(\mathbb{Q}))$ finitely generated? Are there any algorithmic methods (ideally with available software implementations) which can give information about this supposed group in explicit examples?