On finite groups in which the twisted conjugacy classes of the unit element are subgroups

We consider groups G such that the set G, =\g^{-1g^|g G\} is a subgroup for every automorphism of G, and we prove that there exists such a group G that is finite and nilpotent of class n for every n N. Then there exists an infinite nonnilpotent group with the above property and the conjecture 18. 14 of 5 is false.