Is it possible to create a contact list that is synchronized with a specific permissions group? The goal of this is to have a shared contacts list so that everyone within a permissions group has everyone else's contact information and can then sync the list with their outlook if they want.
Out of the box this is not possible to create such a list. The use case of a contact list is more to store additional accounts related to a site or project. For the contact internal contact information you can use a exchange distribution group.
The site user web part is capable to show user from a specific SharePoint group on the web page.
You can also email enable SharePoint user groups