Connected components of an orbit generated by Lie group action

Question

I have a matrix representation of a Lie group $G$ acting on $\mathbb{R}^n$. Given a point $x\in\mathbb{R}^n$, the group generates an orbit $G.x$. My question is about the connectedness of $G.x$. Let $(G.x)^\circ$ be the connected part of $G.x$ that contains $x$. Let $G^\circ$ be the set wise stabilizer of $(G.x)^\circ$, i.e. $G^\circ.(G.x)^\circ=(G.x)^\circ$. In that case, can we always define a quotient $G/G^\circ$ such that under its action one connected part changes to another?