This totally nerd-sniped me.
For the role of derivations in automorphisms of group extensions, check out the Wells exact sequence (and more recent work on it e.g. Buckley, Malfait, Jin-Liu), nicely exposited (w/ refs) by Jill Dietz in https://doi.org/10.1017/CBO9781316227343.013.
I had some thoughts about building the McGee graph from covering graphs (or etale covers, at least), but I think maybe they are too complicated to be worth writing down.