Using the Hatcher's notation on proposition 1.39. Let $G(\tilde{X}):=\{ \varphi: \tilde{X} \to \tilde{X} \; \mathrm{Isomorphism} \}$ be the Deck transformation group for the covering space $p: \tilde{X} \to X$.
Question: Could someone please explain to me why $$G(\tilde{X})\cong \pi_1(X,x_0)/p_*(\pi_1(\tilde{X},\tilde{x_0}))$$ holds just when the covering $\tilde{X}$ is a normal covering? Please provide a counterexample when this does not hold when the covering is not normal.
Many thanks.