Currently, We change the name of the team to the managed group when connecting an Existing team to the group. It would be preferred to use the existing team name when connecting it to a managed group. 

Adding one more use case to this feature request:

Please introduce a way to edit the Team name after synchronization with Jira group? 

After synchronization, it is not possible to edit the team name anymore, which makes it impossible to have a friendly name for the team because Jira groups are often synchronized with Entra ID or Active Directory and have peculiar names.