I am an algebraic topologist and am trying to understand some computations related to p-adic complex K-theory and equivariant K-theory. However this has led me into the work of algebraic geometry over $\mathbb{Z}_p$ and I need some help with some basic computations.

I would like to understand the moduli stack of G-torsors over the scheme underlying the multiplicative group $\mathbb{G}_m$ where we are working with schemes over the p-adic integers $\mathbb{Z}_p$ and $G$ is a finite group. I would even be happy to know the cases where $G = \mathbb{Z}/p$ and $G= \mathbb{Z}/q$ with $q \neq p$ are cyclic groups.

Sometimes it is useful to use the heuristic that $\mathbb{G}_m$ is like the circle $S^1 = \mathbb{R}/\mathbb{Z}$. For the circle the stack of G-torsors looks like $[G/G]$ for the conjugation action. Perhaps the $G$-torsors on $\mathbb{G}_m$ hava a similar description? A simple description of this type would be very useful to me. I need this for explicit calculations and so certainly need the description to be as explicit as possible.

A good start would even be computing the set of isomorphism classes of $G$-torsors over $\mathbb{G}_m$. I tried to compute this by finding some nice cover of $\mathbb{G}_m$, but it is getting late here and I haven't found something that works yet.

I have tried googling around and have come up short. Most of the stuff I have found is either about $\mathbb{G}_m$-torsors, or about $G$-torsors for groups related to $\mathbb{G}_m$, or only work over a field.

If this question is too basic, I would be happy even for a reference with some examples carefully worked out so I can learn explicitly how to do this kind of calculation.

I am an algebraic topologist and am trying to understand some computations related to p-adic complex K-theory and equivariant K-theory. However this has led me into the world of algebraic geometry over $\mathbb{Z}_p$ and I need some help with some basic computations.

I would like to understand the moduli stack of G-torsors over the scheme underlying the multiplicative group $\mathbb{G}_m$ where we are working with schemes over the p-adic integers $\mathbb{Z}_p$ and $G$ is a finite group (which we also view as a group scheme over $\mathbb{Z}_p$). I would even be happy to know the cases where $G = \mathbb{Z}/p$ and $G= \mathbb{Z}/q$ with $q \neq p$, i.e., for cyclic groups.

To clarify, for a scheme S defined over the p-adic integers $\mathbb{Z}_p$, the S-points of the stack in question form a groupoid, the groupoid whose objects are S-families of G-torsors over the p-adic multiplicative group $\mathbb{G}_m$. In other words the objects are G-torsors over $$S \times_{spec \; \mathbb{Z}_p} \mathbb{G}_m.$$ The morphisms are the obvious ones, and everything is defined over the base ring $\mathbb{Z}_p$, which should probably be viewed as either a topolological ring or a pro-ring/ind-scheme. I am willing to be quite flexible on this point, as well as the particular topology used (Zariski, etale, flat, etc).

Sometimes it is useful to use the heuristic that $\mathbb{G}_m$ is like the circle $S^1 = \mathbb{R}/\mathbb{Z}$. For the circle the stack of G-torsors looks like the quotient stack $[G/G]$ for the conjugation action. Perhaps the $G$-torsors on $\mathbb{G}_m$ hava a similar description? A simple description of this type would be very useful to me, i.e., as a quotient stack or really any presentation at all. I am not very picky, but I need this for explicit calculations and so certainly need the description to be as explicit as possible.

A good start would be computing the groupoid of G-torsors over $\mathbb{G}_m$ (ignoring S-families) or even the set of isomorphism classes of $G$-torsors over $\mathbb{G}_m$. I tried to compute this by finding some nice cover of $\mathbb{G}_m$, but it is getting late here and I haven't found something that works yet.

I have tried googling around and have come up short. Most of the stuff I have found is either about $\mathbb{G}_m$-torsors, or about $G$-torsors for groups related to $\mathbb{G}_m$, or work which only works over a field of some sort or another. To clarify I am interested in $\mathbb{G}_m$ over $\mathbb{Z}_p$, so not over a field, in particular not over a field which has pth roots of unity. However I would be interested in the answer over those other rings in-so-far as they help get me the answer over $\mathbb{Z}_p$.

If this question is too basic, I would be happy even for a reference with some similar examples carefully worked out so I can learn explicitly how to do this kind of calculation.