Show that if $\alpha : G \rightarrow G'$ is a homomorphism and $\alpha$ is injective, then the induced map $\bar{\alpha} : G \rightarrow \text{Im}(\alpha)$ is an isomorphism, where $\text{Im}(\alpha) = \{ \alpha(g) : g \in G \}$.
I have already shown that $\text{Im}(\alpha)$ is a subgroup of $G'$. However, I'm not sure how to show that the induced map is an isomorphism.
The induced map is very clearly an isomorphism as you have assumed that the original map is an isomorphism (these two maps are the exact same map if the original map is an isomorphism). Are you sure this assumption is correct?