Let $G$ be a compact Lie group. Let $\mathcal{A}$ be the space of connections on the principal trivial $G$-bundle $G\times \mathbb{R}^4$ possibly with some growth condition (to specify it is a part of the question).
ADD: Let us comment on the growth condition. Let $\nabla^0$ be the trivial connection on $G\times X$. Any connection $\nabla$ has the form $\nabla=\nabla^0+A$ where $A$ is the section of the bundle $\Omega^1\otimes Lie(G)$ of 1-forms on $\mathbb{R}^4$ with values in $Lie(G)$. The growth condition should be imposed on $A$.
The gauge group $\mathcal{G}:=Maps(\mathbb{R}^4\to G)$ acts on $\mathcal{A}$ in the usual ways.
Can the action of $\mathcal{G}$ on $\mathcal{A}$ be free? E.g. for $G=SU(2)$? If not, is it true that the set of connections with non-trivial stabilizers (or infinitesimal stabilizers) is 'very small' in some sense?
Remark. If $G=U(1)$ then the action of $\mathcal{G}$ on $\mathcal{A}$ is free provided we impose a growth condition on connections such that they should vanish at infinity at least along a given direction.
