There are three locations for fonts:
macOS System Fonts (you can't touch these)
/System/Library/Fonts
All user fonts
/Library/Fonts
Each user fonts
/Users/<user>/Library/Fonts
In Font Book Settings -> Sidebar you can enable showing all three.
In the Font Book Settings -> Installation you can install by default to Current User or All Users or any Libraries you created in Font Book.
I suspect the 3rd Party font management tool created a Library and that is set as the default installation location. You'll want to move the fonts to Current User (My Fonts). I think your terminology usage of "Group" would actually be a Library.
Collections are more like tags in that it filters the list to show members of the collection but adding or removing to or from a collection or a smart collection doesn't move the actual font in My Fonts, All User Fonts, or any Library.
In Font Book Settings -> Advanced there's a checkbox for "Show fonts managed by non-Apple utilities" which results in adding another font list called "Third-party Managed Fonts".
Click the Help menu in Font Book, it has pretty decent documentation that explains a great deal.