This is a question related to something I saw in the book Confoliations by Thurston and Eliashberg. Consider three G-bundles $M \rightarrow F$ and two dimensional plane fields $\xi$:
A. $M= F\times \mathbb{R}_+$, $G=Diff_0\mathbb{R}$ germ of diffeos at the origin of diffeos $\mathbb{R}_+ \rightarrow \mathbb{R}_+$ and $\xi$ is the germ of a plane field tangent to the zero section.
B. $M= F \times \mathbb{R}$, $G= Diff_{c}\mathbb{R}$ compactly supported diffeos and $\xi$ is the product foliation outside a compact set
C. $M$ has fiber $S^1$, $G=Diff_+(S^1)$ of orientation preserving diffeos of $S^1$ and $\xi$ is just transverse to the fibers.
$\xi$ defines an Ehresmann connection for these bundles for sure but in the book it is written that it is a G-connection. For this to be, lift of $\xi$ to $M$ needs to be invariant under the action of chosen $G$. So does the statement meant to say that given any $\xi$ there always exists some $G$ for which $\xi$ is a G-connection? How does one show this?
Thanks