Example

The kernel K of L need not be trivial:
If G = Z and H = 3Z, then the kernel is equal to 3Z.

For, L_n(m + 3Z) = (n + m) + 3Z describes the action on the cosets and it is clear that L_n(m + 3Z) = m + 3Z holds for all m if and only if n 3Z. Thus, K = H.

It is true, however, that the kernel is always a subgroup of H. Do you see why?