I gave two lectures on this stuff to a new research seminar on Groups and Graphs, run by Vijayakumar Ambat in Kochi, Kerala. The first was an introduction to the hierarchy, the second was about cographs and twin reduction, why they matter for this business, and questions like “For which groups is the power graph a cograph?”
The lectures were recorded; they are on YouTube here and here.
Peter Cameron's Blog 
Thanks for initiating a discussion on G and G. I wish that some collaborative works will be the outcome.
Thank you Vijay for setting it up. It was a very good experience for me. As I said, I especially liked the way people would interrupt with questions and comments. This is exactly what my students don’t do!