I'm trying to understand the Weil restriction of a scheme (since I'm reading a paper which uses it). I'm even having troubles trying understanding the following "toy" example.
Toy example. Let $X$ be a nice variety over a field $k$ (I can assume $k$ has characteristic 0 if it makes things easier). Let $Y\to X$ be an $F$-torsor, where $F$ is a finite linear algebraic $k$-group. Let $Z \to Y$ be a $G= \mathbb{G}_{m,k}$-torsor over $Y$. We can see $Z \to Y$ as a $G_Y = G \times_k Y$-torsor over $Y$. Now take the Weil restriction $R=R_{Y/X}(G_Y)$. This is an $X$-group scheme and turns $R_{Y/X}(Z) \to X $ into an $R$-torsor over $X$.
I have some questions about the above toy example.
What is $R$ explicitly? I know it's a group scheme, but can we say more? How does it relate to the original $G= \mathbb{G}_{m,k}$? Do we have any morphism $R \to G_Y$? Do we have a morphism $R_{Y/X}(Z) \to Z$?
Any detailed comments or hints, suggestions, constructions for any of the above questions would be really helpful, as I really don't understand this Weil restriction business. I've tried reading chap 7.6 of the book on Neron models by Bosch etc, but it doesn't really help my understanding of even the above toy example. In general, I don't know what sort of objects $R$ is, since the definition of Weil restriction is too abstract.
Edit. Is $R$ just $\mathbb{G}_{m,k}^d \times_k X$ where $d$ is the cardinality of $F(\bar{k})$?