Not each subgroup $H\subset G$ can be the kernel of a homomorphism

Question

My book says that each subgroup $H\subset G$ can be the image of a homomorphism; consider the inclusion mapping $f\colon H\to G\colon x\mapsto x.$ However, not each subgroup can be the kernel of a homomorphism.

I don't see why this should be a problem. Consider $H\subset G$ a subgroup. We know that $H$ is not empty. Now consider the mapping $f\colon H\to G:x\mapsto e$. This way, $H$ is the kernel of $f$, and $f$ is a homomorphism, because $f(a+b)=e=e+e=f(a)+f(b)$.

So why did they say that not each subgroup can be the kernel of a homomorphism?

Answer 1

What is meant is that there does not exist a homomorphism $f:G\rightarrow K$ such that $\text{ker } f = H$. In fact, such an $f$ exists iff $H$ is normal. To see this, note that the kernel of a homomorphism is a normal subgroup. Conversely, if $H$ is normal, one can take the canonical projection $G\rightarrow G/H$.