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

We consider groups G such that the set \([G,\varphi ]=\{g^{-1}g^{\varphi }|g\in G\}\) $[G,\phi ]=\{g^{-1}{g}^{\phi }|g\in G\}$ is a subgroup for every automorphism \(\varphi \) $\phi$ of G, and we prove that there exists such a group G that is finite and nilpotent of class n for every \(n\in \mathbb N\) $n\in \mathbb{N}$ . 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.