If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a finitely generated and possibly infinite abelian group.

This question concerns a similar situation for étale group schemes over a field, and it came up while thinking about Serre's method of specialization for Mordell-Weil groups via Hilbert irreducibility.

Suppose $G$ is a commutative *étale* group scheme over $\mathbf{F}_q$ (I am not assuming $G$ is of finite type. Only *locally* of finite type).

Is there an example of such $G$ such that $G(\mathbf{F}_q)$ is finite, or at least torsion, and $G(\mathbf{F}_q(t))$ is finitely generated and with a point of infinite order?

Suppose now that $f: G_1\to G_2$ is a surjective map of commutative étale group schemes over $\overline{\mathbf{F}}_q$ (not necessarily of finite type). Assume that $G_2(\overline{\mathbf{F}}_q)$ is a *finitely generated free* abelian group.

Is there a homomorphism of commutative étale group schemes over $\overline{\mathbf{F}}_q$, $g : G_2\to G_1$, such that $f\circ g$ is the identity on $G_2$?

In other words, a section to $f$ that is a homomorphism.