Let $(C,c_0),(X,x_0)$ be topological spaces.
Let $p:(C,c_0)\rightarrow (X,x_0)$ be a covering map.
Let $p_*:\pi_1(C,c_0)\rightarrow \pi_1(X,x_0)$ be the induced homomorphism by $p$.
Then, is it true that $p_*(\pi_1(C,c_0))$ is a normal subgroup of $\pi_1(X,x_0)$?
Let $\phi:\pi_1(X,x_0)\rightarrow p^{-1}(x_0)$ be the lift correspondence of $p$.
Define $H=p_*(\pi_1(C,c_0))$.
Let $\alpha,\gamma$ be loops at $x_0$.
It's a theorem in Munkres-Tolology that: $[\alpha]\in H[\gamma]$ iff $\phi([\alpha])=\phi([\gamma])$.
Doesn't this imply that $H$ is a normal subgroup of $\pi_1(X,x_0)$?