On groups where the twisted conjugacy class of the unit element is a subgroup

We study groups $G$ where the $\varphi$-conjugacy class $[e]_{\varphi}=\{g^{-1}\varphi(g)~|~g\in G\}$ of the unit element is a subgroup of $G$ for every automorphism $\varphi$ of $G$. If $G$ has $n$ generators, then we prove that the $k$-th member of the lower central series has a finite verbal width bounded in terms of $n,k$. Moreover, we prove that if such group $G$ satisfies the descending chain condition for normal subgroups, then $G$ is nilpotent. Finally, if $G$ is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of $G$.