Are kernels unique to homomorphisms?

No, a homomorphism is not uniquely determined by its kernel. Consider the following two homomorphisms from $\mathbb{Z}_2$ to $\mathbb{Z}_2\times\mathbb{Z}_2$: one sending $1$ to $(0,1)$ and the other sending $1$ to $(1,0)$. They're both homomorphisms with the same kernel to the same group, but they are different homomorphisms.

Unless $G$ has a "simple" structure, there are many isomorphisms $:G \to G$ and they all have a kernel of $\{e_G\}$.

The identity homomorphism and the homomorphism $n\mapsto -n$ from $\mathbb{Z}$ to $\mathbb{Z}$ have ...?... kernel.