For a group homomorphism, does there exist a subgroup of the domain isomorphic to the image? Let $\phi : G \to H$ be a group homomorphism. Is is it true that, there exists a subgroup $K$ of $G$ such that $K$ is isomorphic to the image of $\phi$?

Edit: How about for finite $G$?
HINT: $\Bbb Z$ has finite homomorphic images of all sizes; does it have finite subgroups of all sizes?
Yeah. If there is a homomorphism $\phi$ from $G$ to $H$, then there is a isomorphism from $G/\ker {\phi}$ to $\phi(G)$
(namely, from $a\ker\phi$ to $\phi(a)$ is the isomorphism). As $\ker{\phi}$ is a normal subgroup of $G$ it is evident that its factor group is a subgroup.