Salesforce Chatter Groups may be created internally within a Salesforce org or in the context of a site.
If a Chatter Group is created within an org internally, only internal users may be added to that group. If a Chatter group is created within a site, then users may be added to the group only if they are members of the site.
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.
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.
Idea Exchange - Enable Chatter Groups between Main Org & Communities
000389234

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