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

Abstract We consider groups G such that the set G, =\g^{-1g^ |g G\} G, φ = g - 1 g φ | g ∈ G is a subgroup for every automorphism φ of G, and we prove that there exists such a group G that is finite and nilpotent of class n for every n N n ∈ N. Then there exists an infinite not nilpotent group with the above property and the Conjecture 18. 14 of Khukhro and Mazurov (The Kourovka Notebook No. 20, 2022) is false.