Hatcher's Algebraic Topology Exercise 1.3.28 is :
Exercise 1.3.28. Show that for a covering space action of a group $G$ on a simply-connected space $Y$, $\pi_1(Y/G)$ is isomorphic to $G$. [If $Y$ is locally is path-connected, this is a special case of part (b) of Proposition 1.40.]
Proposition 1.40 is :
Proposition 1.40. Given a covering space action of a group $G$ on a space $Y$,
(a) The quotient map $p : Y \to Y/G$ is a normal covering space.
(b) $G$ is the group of deck transformations of the covering space if $Y$ is path-connected.
(c) $G$ is isomorphic to $\pi_1(Y/G)/p_*(\pi_1(Y))$ if $Y$ is path-connected and locally path-connected.
I think Exercise 1.3.28 is a special case of part (c) of Proposition 1.40, not (b). But there is no mention in the errata list in Hatcher's homepage : http://pi.math.cornell.edu/~hatcher/ . Am I wrong?
This community wiki solution is intended to clear the question from the unanswered queue.
You are right. Part (b) does not allow to conlude that $G \approx \pi_1(Y/G)$.