I am looking to track the setting change that a group owner can make on their groups, changing the radio button selection of "who can edit the membership of the group?" from "Group Owner" to "Group Members"?
I think in your case you want to know who did that historically? Unless you have the audit settings correctly established I don't think you will be able to do that.
To make sure you are good for the future:
- Navigate to the desired site
- In the site settings, make sure that Reporting is on.
- Site Settings Audit Logs Reports, than security settings
You will need to filter and navigate the excel sheet. But there you should be able to see all the security changes that were made.
Can you check under Site Settings -> Site Collection Audit Settings -> Editing users and permissions.
Let me know if that is ticked. Than try doing that with another security group what you described in your message and see if that is captured.
Hope that helps.