We have an article that explains what is possible when working with Outlook 2010 and SharePoint 2010:

Synchronize SharePoint 2010 content with Outlook 2010

Unfortunately it does not explain what is not possible but Outlook offers this to the user.

Imagine the following scenario:
You have a Contact List in SharePoint.
You clicked on “Connect to Outlook”.
Outlook is now able to synchronize the contacts in both directions.
Click in Outlook on “New Contact Group”.
Enter a Name for that group.
Click on “Save & Close” and you will get an error message that could confuse you:

"Your changes cannot be saved because you don't have permission to modify some or all of the items in this folder. Do you want to save a copy of this item in the default folder for the item?"

In short: It is not possible to also save “Contact Groups” in the SharePoint Contact List.

A bit longer: The SharePoint Contact List has some columns as all the other Lists are having. You could also compare the Contact List with your own customized List. Storing Contacts is the main reason for that List but a Contact Group would have a complete different Schema. Edited 15/Mar/2012. In general it would be possible to use more than one content type (can be also called as Schema) in a List but that would need another content type on the Contact List and also Outlook would need to know that. In question might be how that could look and how the user experience will be reduced when working with the List and not only with Outlook.

SharePoint is the Server and offers some services but the Consumer of those services is responsible for the own offered services and error messages.