  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 $G$.
