Let $l/k$ be a finite extension of fields. Let $G$ be an affine $k$-groups scheme. Let $G_l = G \times_k l$. Is it true that $\mathfrak{R}_{l/k}(G_l) = G$, where $\mathfrak{R}_{l/k}(-)$ is the restriction of scalars?
This appears to be false. So my new question is:
Is it true that $\mathfrak{R}_{l/k}(G_l) = G^{[l:k]}$?
What does "=" mean here, and do you have any examples of group schemes where this holds for some nontrivial extension?
If $k = \mathbb{R}$, $l=\mathbb{C}$, and $G=\operatorname{Spec} \mathbb{C}$, then certainly something goes wrong, as $G\times_k l$ is isomorphic to two copies of $G$. This is a counterexample to every interpretation I can think of to the problem, but it's possible that I'm overlooking something.