Salesforce Internal Chatter Groups and Experience Site Chatter Groups

Chatter Groups created internally cannot be viewed in a site which goes the same with those Chatter Groups created in the community which aren't visible inside their Salesforce organization.

This separation exists because internal Chatter Groups are scoped to the Salesforce org's user base, while Experience Site Chatter Groups are scoped to site members only. These are separate data contexts by design. Currently, there is no native configuration to make a group visible in both the Salesforce org and an Experience Site simultaneously.

As of this moment, this is working as designed and there is no existing option to enable Chatter Groups to be visible both in the Salesforce organization and a Digital Experience Site.