Hi Jim,
That group, and the Lead Service Line Inventory Content group, are automatically created and deployed when a solution includes an ArcGIS Hubsite. From testing on my end, it seems you can only add users to the Core group if they are a Creator user type or higher. The Content group doesn't have a minimum requirement. Both groups are not part of the solution even though they are deployed with the solution (as I mentioned above they are system generated because of the Hub Site).
Are you finding the same issue with the other Lead Service Line Inventory groups such as Lead Service Line Editors or Lead Service Line Office Viewers?