Is it possible to keep a SharePoint group in sync with the an outlook group? So that owner of the site doesn't have to make changes in both places?
As a short answer: Unfortunately , it's not possible ,
You can sync contact,calendar and task list with outlook but there is no way to sync groups that because of There is a major design and functionalty difference between SharePoint Group and the People contacts group in outlook.
Also if you cheked the defination of each group below, you will know why you can't perform group sync !
Outlook Contact Groupis grouping of email addresses collected under one name. (Note: you can't create a Contact Group at contact list synced from SharePoint Contact list that due to design too for more details check contact group sync error)
SharePoint Groupis a collection of users that have specific permission role like Contribute role , the Read role Design role, Full Control role ..etc.