In exercise 1.3.28 in Hatcher's Algebraic Topology, we are asked to show that for a covering space action of a group $G$ on a simply-connected space $Y$, $\pi_1(Y/G)$ is isomorphic to $G$. This is a special case of Proposition 1.40(c).
My question is: What happens if $Y$ is not locally path-connected and not simply connected? Is there a counterexample to show that Proposition 1.40(c) fails? I am still assuming $Y$ is path-connected.
Edit: I should clarify that I want to find a counterexample to proposition 1.40(c) if $Y$ is not locally path-connected, which means that I want to find a a covering space action of a group $G$ on a path-connected space $Y$ such that $G \ncong \pi_1(Y/G)/p_*(\pi_1(Y))$, where $p: Y \to Y/G$ is the quotient map. Proposition 1.40(c) implies that such a $Y$ cannot be locally path-connected, while Exercise 1.3.28 tells us that such a $Y$ cannot be simply connected.