You will need a Global Admin account to perform this task.
Connect to Exchange Online via PowerShell.
Initially add desired email address to the group:
1 | Set-UnifiedGroup -Identity "Group Name" -EmailAddresses:@{Add ="desiredemail@domain.com"} |
Then set that email address as primary:
1 | Set-UnifiedGroup -Identity "Group Name" -PrimarySmtpAddress "desiredemail@domain.com" |