I wish to delete my accepted answer here. When I flagged it for mod attention, the review was that there was there was no evidence to support my contention. I do not know what to do at this stage. There seems to be no way to contact a moderator directly without flagging.
For the record the $f$ in my answer is not a homomorphism, and in fact I believe that I have misunderstood the question ($\{0,3,6\}$ is a subgroup of $\mathbb{Z}_n$ where 2 elements are "even" and 1 is odd.)
For the general question:
You should try to convince the OP that you gave the wrong answer. This can be done by adding comments or editing your answer to say explicitly what is wrong.
Once the OP unaccepts your answer you can delete it yourself.
For the specific answer you linked to:
Why do you want to delete? I think your proof works. You have the homomorphism $\mathbb{Z}_{2k} \to \mathbb{Z}_2$ since "even + even" and "odd + odd" are both even, while "even + odd" is odd. Since $H\subseteq \mathbb{Z}_{2k}$ is a subgroup the homomorphism descends to $H$ with no problem.
Note that $\{0,3,6\}$ is only a subgroup of $\mathbb{Z}_9$ and $9$ is not even.
