Hi @sobermeyer Box
As of this moment, there is no native integration between Google Groups and Box that automatically syncs members between the two, including adding members to Box folder/s. This is a limitation of the Google IdP integration with Box, as it is managed by Google.
I suggest to check out this article from Google: Configure Box Enterprise auto-provisioning, which explains how to set up automated user provisioning (auto-provisioning) so that any changes you make to user accounts in Google Workspace are automatically synced with third-party apps.
For further assistance regarding this integration, or see if they have custom app built for this approach, kindly reach out to Google.
Furthermore, for assistance with Box account-specific concerns or other issue with integration, please submit a support ticket so our Product Support team can investigate. Hope this helps!