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

We consider groups $G$ such that the set $[G,\varphi]=\{g^{-1}g^{\varphi}|g\in G\}$ is a subgroup for every automorphism $\varphi$ of $G$, and we prove that there exists such a group $G$ that is finite and nilpotent of class $n$ for every $n\in\mathbb N$. Then there exists an infinite nonnilpotent group with the above property and the conjecture 18.14 of $[5]$ is false.