Let $k$ be an algebraically closed field.
We assume all schemes we consider are $k$-schemes.
Let $P \rightarrow X$ be a principal $G$-bundle, where $G$ is an algebraic group.
We assume $X$ is affine $(X := \text{Spec}(A))$.
Let $X \rightarrow X' := \text{Spec}(A')$ be a thickening.
Note that thickening means $X \rightarrow X'$ is a closed immersion and its ideal sheaf is nilpotent.
Then, do we have a lifting of the principal bundle $P$ ?
I.e., We have the following cartesian diagram
$\require{AMScd}$
\begin{CD}
P @>>> P'\\
@VVV @VVV\\ X
@>>> X'
\end{CD}
, where $P' \rightarrow X'$ is also a principal $G$-bundle ?
Edit :
I originally came across this problem in the following question.
Let $Y$ be a $k$-scheme with a $G$-action and $P \rightarrow Y$ be a $G$-equivariant morphism ($P$ is defined as above).
Then, classify the liftings of $P \rightarrow Y$ to $P' \rightarrow Y$ s.t. they are also $G$-equivariant if a lifting $P' \rightarrow X'$ of a principal $G$-bundle $P \rightarrow X$ with respect $X \rightarrow X'$ exists.
Best Answer
This follows from the infinitesimal lifting criterion for smoothness: An algebraic stack $\mathcal{X}$ locally of finite type is smooth if and only if for every nilpotent thickening $S \to S'$ of affine schemes, any map $S \to \mathcal{X}$ extends to $S'$. If $G$ is an algebraic group, its classifying space $BG := [pt/G]$ is an algebraic stack locally of finite type for the etale topology. This is so simply because $BG$ has a smooth cover by a point.
I'm not sure what the best reference is for the infinitesimal lifting criterion, but you can see one direction of the proof on page $36$ here.