I was trying to do it the way that requires less actions from Pete, since he's busy.
As far as I understand, the message "forked from petehunt/PyMySQL" can be removed in two ways:
A) detaching the repository from the network;
B) making the repository the root of the network.
In the first case, that message would still appear on other forks. In the second case, the other forks (and even petehunt/PyMySQL) would have the message "forked from PyMySQL/PyMySQL".
Both options would require the intervention of GitHub's staff, and option B would also require that Pete send an email to
sup...@github.com requesting to make PyMySQL/PyMySQL the root of the network (it needs to come from him because he is the owner of the current root).
A third option is to move back the fork to my user account and ask Pete to move his repo to the organization, as you suggest. I think that most PRs there conflicts with the code ported to Python 3, and most issues were solved by the port. But sure it would be nice to have the PRs/issues closed accordingly and kept for future reference.
I would like to hear more opinions on all that.