Readit NewsThe easy solution here would be to just have two different names: (general) automorphisms (of which there might be many) and automorphisms-that-keep-R-fixed (of which there are just the two mentioned.
If you make this distinction, then the approach of construction of C should not matter, as they are all equivalent?
Cory Doctorow's wonderful sci-fi book "Down and Out in the Magic Kingdom" explored exactly this in interesting ways. In the book people in the future can live essentially forever by transferring their consciousness into new bodies. They can also back up the contents of their consciousness, something most people do nightly but certainly before doing some dangerous extreme sport. Doing dangerous things without backing yourself up is considered tantamount to suicide since you lose all the memories and personal growth, essentially the person you became since your last backup.
People do get bored and will sometimes choose to "deadhead" for hundreds of years at a time, which is putting yourself into stasis and skipping those centuries. The book is full of provocative ideas about how practical immortality might actually work on a personal and societal level.
Note that Weizenbaum was an AI critic: Weizenbaum's intention was not for Eliza to pass the Turing test, but to show to people that a clearly not intelligent program based on primitive pattern matching can appear to behave intelligently.
He failed: His own secretary wanted to be left alone with the software and typed in her personal problems. Work on Eliza (1963-65, paper published 1966) until today is mostly misunderstood.
In my opinion, the gap in performance is less important, but the commercial offerings are typically more robust/reliable.
Similarly, researches who do spend their time implementing solver algorithms and running tedious computational experiements (the work that the software vendors put in) have historically had difficulties getting academic credit for their work, because the journals favored theoretical work.
That being said, with HiGHS and SCIP, we have two open-source solvers developed in an academic setting, with a lot their graduates joining commercial software vendors. So it's not like these are two completely separate worlds.
For a nice entry game for a group setting, I recommend Carcassonne. It has a simple and engaging basic gameplay with a surprising amount of depth, that can easily be scaled up and down in complexity depending on your group's preference and experience level by simply adding more pieces/mechanics.
Our oldest child is now capable of the base game, and I can still make it interesting for me by going for secondary objectives, such as filling difficult gaps ;-)