Lie-group existence on universal covering manifold

Question

Let $X$ be an n-dimensional smooth manifold with Lie group $G$ acting transitively on $X$, i.e. $X$ is a homogeneous space. Let $\tilde{X}$ be the associated universal covering space. To what extent is it known that there exists a Lie group $\tilde{G}$ on $\tilde{X}$ and a surjective group homomorphism $\psi:\tilde{G} \rightarrow G$ such that the covering map $\varphi$ is equivariant with respect to this homomorphism, i.e. that \begin{equation} \varphi(\tilde{g}(\tilde{x})) = \psi (\tilde{g}) \varphi (\tilde{x}), \end{equation} where $\tilde{x}\in \tilde{X}$?