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

We consider groups G such that the set [G,φ]=⁻¹gφ|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. Then there exists an infinite nonnilpotent group with the above property and the conjecture 18.14 of $[5]$ is false.