[SharePoint Online] sites can use [Azure AD] security groups to manage user access and permissions. Noteworthy, these security groups can be applied directly to sites, lists, and/ or items. And every security group used on the SPO site is saved to the site member list. This member list roughly functions as a site cache and records every AAD object ever bound to the site:
- https://<tenant name>.sharepoint.com/_layouts/15/people.aspx?MembershipGroupId=0
Though, when security groups are renamed, the site cache retains the original name. For these updates to be reflected in SPO, the groups must first be deleted from the site collection, then reapplied to the SPO target.

Conclusion:
There is a manual, not automatic process for Azure AD security group renames to be reflected in SharePoint Online…
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