Renaming an AAD group would not have any impact on existing permissions and everything should be same.
Based on my experience, the name change is not reflected automatically in user details within Azure SQL database/synapse.
Still get the old names with the below query:
SELECT r.name role_principal_name, m.name AS member_principal_name
FROM sys.database_role_members rm
JOIN sys.database_principals r
ON rm.role_principal_id = r.principal_id
JOIN sys.database_principals m
ON rm.member_principal_id = m.principal_id
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