You should definitely find a Git server. It may be Github, or Bitbucket, or one of dozens of others. There are also solutions that you can run onsite (not under z/OS however that I'm aware of).
You will also need the Rocket Git port to work with the z/OS datasets unless you want to copy the z/OS data elsewhere.
For working on z/OS check out
ZIGI - the z/OS ISPF Git Interface. It is an open source project hosted on Github.
If you have any questions let me know via a reply or private message.
Full disclosure - I'm one of the primary authors/contributors to ZIGI.
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Lionel Dyck
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