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Oct 18, 2016 at 15:20 vote accept Vipul Naik
Oct 17, 2016 at 18:04 comment added YCor Note that it's based on the same idea as your solvable example: one choice of group containing the derived subgroup, assuming that the action on [some nontrivial part of] the abelianization is trivial. The only difference is that in the nilpotent world one need to struggle a little further to find the relevant examples; fortunately the cumbersome computational work had already been done (cf references).
Oct 17, 2016 at 18:01 history answered YCor CC BY-SA 3.0