Consider a sheaf of groups $G$, equipped with a left torsor $P$ and another left action $G$ on some $X$. Form the contracted product $P \times^G X := (P \times X)/\sim$ where $\sim$ is the antidiagonal quotient: $(g.p, x)\sim (p, g.x)$.

**Q1:** When is $P\times^G X$ trivial? I.e., when do we have an isomorphism $P \times^G X \simeq X$?

Partial answer: $P \times^G X \simeq X$ over $[X/G]$ iff $P \times [X/G]$ is a trivial torsor over the stack quotient $[X/G]$.

Proof: We can rewrite $P \times^G X$ as a contracted product of two *torsors* $(P \times [X/G])\times^G_{[X/G]} X$. Then we contract with ``$X^{-1}$'' -- the inverse to contracting with $X$ as a torsor over $[X/G]$ and we win. (as in B. Poonen's Rational Points on Varieties, section 5.12.5.3)

Am I allowed to do this? This argument probably shouldn't have to appeal to algebraic stacks and may be somewhat dubious.

**Q2:** If I have one isomorphism $P \times^G X \simeq X$, can I choose another one that lies over $[X/G]$? Or at least is $G$-equivariant?

**Q3:** Is there a natural way to write the triviality of such a twisted form?

I first thought $P \times^G X \simeq X$ iff $P$ was trivial, which is clearly false for trivial actions on $X$. Then I was excited to have the pullback $* \to BG$ represent triviality of the *twisted form* $P \times^G X$ as well as the torsor $P$. Is there a natural representative of the sheaf of isomorphisms between $P \times^G X$ and $X$?

These can all be sheaves, although I'm primarily interested in $G = GL_n, PGL_n, SL_n$, etc. acting on $X = \mathbb{A}^n, \mathbb{P}^n$ as appropriate. More ambitious is $G = \text{Aut}(X)$ for even simple $X$. I'd be happy with answers in any level of generality.

Due Diligence Statement: I'm a novice in the area of "twisted forms" of varieties, so I apologize if the above is evident or obtuse. I checked all the "similar questions" listed here and couldn't find an answer.