I am stuck on a problem studying for my prelims : Let $G$ be a group, $N \unlhd G$ a normal subgroup and $\alpha \colon G \to G$ an automorphism of $G$ such that $\alpha(N) \leq N$. The first part is to prove that $\alpha$ induces an automorphism $\overline{\alpha}$ of $G / N$. The second part is to find an example where $\alpha$ restricted to $N$ and $\overline{\alpha}$ are both identities but $\alpha$ is not the identity.
I am stuck in the first part... So since $\alpha (N) \subseteq N$, there is an induced endomorphism $\overline{\alpha} \colon G/N \to G/N$, which clearly is surjective. However, I cannot prove the injectivity, which is equivalent to proving that there is an induced hm $\overline{\alpha^{-1}}$, which is the same as $\alpha^{-1} (N) \leq N$.
We would be done if the group $G$ is assume to be finite for example, but I don't see how to prove it general. Thanks for your help