One approach to classifying the coverings of a nice space $X$ without choosing a basepoint is to look at actions of the fundamental groupoid on sets. Another way that seems natural to me is to fix a universal covering $p: \widetilde{X} \to X$ and consider the group $\text{Aut}_X(\widetilde{X})$ of deck transformations of $\widetilde{X}$ in place of the fundamental group.
Here's one thing that confuses me about this: if $X$ is connected and $x \in X$, then $\text{Aut}_X(\widetilde{X})$ acts simply transitively on the fiber $p^{-1}(x)$, and in particular we get an isomorphism $\text{Aut}_X(\widetilde{X}) \cong \pi_1(X,x)$. Since this works for any $x,y \in X$, it seems that we get canonical identifications $\pi_1(X,x) \cong \pi_1(X,y)$, but I thought that this could not be done for nonabelian $\pi_1$.
My other question: how useful is $\text{Aut}_X(\widetilde{X})$ when $X$ is nice but disconnected? Do we have some correspondence between actions of this group and coverings of $X$? It is not clear to me what this group looks for, say, $X = S^1 \coprod S^1$.
Edit: I was, of course, incorrect in my formulation of the first question (thanks countinghaus). But the second question stands: do $\text{Aut}_X(\widetilde{X})$-sets correspond to coverings of $X$ even when $X$ is not connected?
Best Answer
For question one: Just knowing that two groups act simply transitively on the same set set isn't enough to give you a canonical isomorphism between them; you still must pick an element of the set to get the isomorphism. This is equivalent to choosing a path between the two basepoints, which is the old-school way to get a (noncanonical) iso on $\pi_1$s with varying basepoints.