As far as I understand, an automorphism is an isomorphism from a set to itself. If we have a homomorphism $f:M\rightarrow M$, then, from the first isomorphism theoreom, $im(f)$ is a submodule of $M$. As it is injective, the kernel is zero, and hence the image is isomorphic to $M$.
Does this not show that $f$ is an isomorphism from $M$ to itself? What am I missing?