Fibres of a group homomorphism are cosets of its kernel ($\phi (g)=\phi (h)⟺h^{-1}g\in \ker \phi$)

For a group homomorphism $\phi :G\to H$, two inputs give the same output exactly when $\phi (h^{-1}g)=e_H$. Consequently the fibre over an output $\phi (h)$ is the left coset $h\ker \phi$. These cosets identify precisely the inputs indistinguishable to $\phi$, which is the mechanism behind the first isomorphism theorem.