[ Update August 23, 2026: the updated paper looks better now. Lots of stuff previously written has been left but after a few frantic days of working on this, I feel better. Here are the main results
Main results are: all connected Lie groups are natural. An arbitrary non-abelian, finitely generated group is natural
if and only if it is not generalized dicyclic. A finitely generated abelian group is natural if and only if it is either finite of odd order or of the form 
There were seismic shifts in the update and correction of the paper on natural groups. I reflect a bit on this. Updates are currently done here on this document. The notion of “natural group” seems to fit very well with the existing literature on graphical regular representation, especially with the theory of Leemann and de la Salle. Things are not totally direct as natural does not necessarily mean to have a GRR. There is also relation to the notion of iso-groups of Niemiec. There is no direct relation either way. Iso-groups deal which is a notion for topological transformation groups. Naturality is an abstract group or metric notion. A group of homeomorphism on a space X is called an iso group if there exists a metric d on X such that G=Iso(X,d). In our case, we want the space itself to be both the group and the metric space. I’m still working on getting the old paper from 2022 patched. But it appears that most could be streamlined to a 2-3 pages including some quite strong results like a classification for all finitely generated groups or all connected Lie groups. This was unexpected but was made possible by becoming aware of a theory by Leemann and de la Salle. An important role is played by involutions, as this allows to show that groups are natural.
One thing I just notice to be understated in the presentation is the restriction of having at most countably many involutions. This can be extended to at most a continuum of involutions. This so allows application to Lie groups. It does not work in general, if the number of involutions exceed the continuum.
