Send different notification for specific groups

Is there a way that a ticket assigned to a group can have a different notification from the template? For instance, if a ticket is opened by an external email address, they receive a notification that the ticket was opened, but it does not contain a URL to the help desk, while internal email addresses do.

If this isn't possible, can we disable all notifications for tickets opened in a specific group?

If neither of these are possible, can we set it up where a group receives a different URL when they are external vs internal? For instance xyz@internal.com would see a url to helpdesk.internal.com and xyz@external.com would see a url to helpdesk.external.com.