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?