This question is about a statement from Section 3.4 of Thurston's 1997 book [1]. There $X$ denotes a connected real analytic manifold, $G$ denotes a group of real analytic diffeomorphisms acting transitively on $X$, and $p : \widetilde{X} \to X$ denotes the universal cover of $X$.
The following line appears on page 143: There is a covering group $\widetilde{G}$ acting on $\widetilde{X}$, namely the group of homeomorphisms of $\widetilde{X}$ that are lifts of elements of $G$.
I don't see where before this claim topological assumptions on $G$ are stated. In the situation I am currently studying $G$ is a Lie group, so let us assume this for simplicity. What is the usual construction of the topology on $\widetilde{G}$ so that $\widetilde{G}$ is a topological group and $\widetilde{G} \to G$ is a covering map? Does this construction appear in a standard reference? Below the line I include some basic steps that get me up to this point, and then after I include some ideas on how to answer the question.
Following the book, we define $\widetilde{G} := \{ \widetilde{g} \in \mathrm{Homeo}(\widetilde{X}) : \exists g \in G: p \circ \widetilde{g} = g \circ p \}$. If $g_1, g_2 \in G$ satisfy $g_1 \circ p = p \circ \widetilde{g} = g_2 \circ p$ for some $\widetilde{g} \in \widetilde{G}$, then $g_1 = g_2$. So we can define $\rho_{G}$ as the map $\widetilde{G} \to G$ that sends $\widetilde{g}$ to the unique $g \in G$ such that $p \circ \widetilde{g} = g \circ p$. The lifting criterion for covering spaces of [Hatcher, Proposition 1.33] allows one to argue that $\rho_G$ surjects, and then [Hatcher, Proposition 1.33] combined with the unique lifting property for covering spaces of [Hatcher, Proposition 1.34] allows one to argue that $\widetilde{G}$ is closed under the inverse of $\mathrm{Homeo}(\widetilde{X})$. It is also clear from the definition that $\widetilde{G}$ is closed under composition, so that it is a subgroup of $\mathrm{Homeo}(\widetilde{X})$, and that $\rho_G$ is a group homomorphism with kernel equal to the group of deck transformations $\pi_1(X)$ of $\widetilde{X} \to X$.
One idea (applicable at least to the case that $G$ is connected) is to try to argue that $\widetilde{G}$ is a quotient of the universal covering group $\widetilde{\widetilde{G}}$ of $G$, and then topologize $\widetilde{G}$ with the quotient topology. I think we can get a homomorphism $\widetilde{\widetilde{G}} \to \widetilde{G}$ mapping onto what should eventually be the identity component of $\widetilde{G}$; I tried to justify in the answer I've written below. I think the idea leading this paragraph cannot immediately work if $\pi_1(X)$ is non-abelian, however, because $\widetilde{G}$ would be connected and contain $\pi_1(X)$ as a discrete normal subgroup, implying $\pi_1(X)$ must be abelian. Perhaps the right modification is to topologize an extension of the universal covering group of $G$, and then show $\widetilde{G}$ is a quotient of that.
Another idea is to give $\widetilde{G}$ the compact-open topology, though this seems like an arbitrary choice. I think this implies discreteness of $\pi_1(X)$, and I think there is perhaps a natural choice of covering of $G$ by evenly covered neighborhoods for $\rho_G : \widetilde{G} \to G$. It is not clear however that the group isomorphism induced by $\rho_G$ is a homeomorphism with this topology.
[1]: Thurston, William P., [Three-dimensional geometry and topology. Vol. 1. Ed. by Silvio Levy], Princeton Mathematical Series. 35. Princeton, NJ: Princeton University Press. x, 311 p. (1997). ZBL0873.57001.
