An old exam question I found asks
"Let $p:\bar{X}\rightarrow X $ be a covering map, such that $\bar{X}$ is path connected and $\pi_1(X)$ is abelian. Are all covering maps regular covering maps"
Definitions:
$p$ is a covering map, if it is surjective and locally trivializing
$p$ is a regular covering map, if the deck transformation group $\Delta(p)$ acts transitively on the fiber. We don't assume $X$ (or $\bar{X}$) to be locally path connected.
I got a proof if $X$ is locally path connected (using lifts), but how would I prove it without that condition? (Or is there a counterexample?)