Étale group schemes and specialization

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.

If $$k \to \ell$$ is any regular field extension (i.e. $$k$$ is algebraically closed in $$\ell$$), then $$X(k) \to X(\ell)$$ is a bijection when $$X \to \operatorname{Spec} k$$ is étale. Indeed, it suffices to show this when $$X = \operatorname{Spec} K$$ for some finite separable field extension $$k \to K$$, and then the statement is that the natural map $$\operatorname{Hom}_k(K,k) \to \operatorname{Hom}_k(K,\ell)$$ is a bijection. If $$k \to K$$ is an isomorphism, then both sides are a singleton, and otherwise both sides are empty by assumption.
Note that $$k \to \ell$$ is regular if and only if $$\ell$$ is a geometrically integral $$k$$-algebra [Tags 037Q and 030W]. In particular, this holds when $$\ell = k(X)$$ is the function field of a geometrically integral $$k$$-variety $$X$$. Notably, $$k \to k(t)$$ (in this case regularity is easily checked by hand).
(A previous version of this answer contained a more geometric argument when $$\ell = k(C)$$ for a smooth geometrically integral curve with a rational point, but neither dimension $$1$$ nor the rational point is needed, and the current proof is actually easier.)
For the second question, recall that the category of finite étale $$k$$-schemes is equivalent to the category of finite sets with a continuous $$\Gamma=\operatorname{Gal}(\bar k/k)$$-action. Taking disjoint unions gives an equivalence between étale $$k$$-schemes and discrete sets with a continuous $$\Gamma$$-action. The group objects are therefore discrete groups with a continuous $$\Gamma$$-action, i.e. groups $$A$$ with a group homomorphism $$\phi \colon \Gamma \to \operatorname{Aut}(A)$$ such that for every $$a \in A$$, the stabiliser $$\Gamma_a$$ is open (equivalently, closed of finite index).
This gives the required counterexample: let $$U \subseteq \Gamma$$ be a subgroup of index $$2$$ (when $$k = \mathbf F_q$$ there is a unique such subgroup), and write $$H$$ for the quotient $$\Gamma/U$$, with nontrivial element $$\sigma$$. Now consider the augmentation $$\mathbf Z[H] \twoheadrightarrow \mathbf Z$$ of $$H$$-modules. The quotient is the constant étale scheme $$\mathbf Z$$ corresponding to the trivial $$\Gamma$$-action. But the surjection is not split as the only invariant elements in $$\mathbf Z[H]$$ are multiples of $$\sigma + 1$$, which maps to $$2$$ under the augmentation.