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

We study groups G where the -conjugacy class of the unit element is a subgroup of G for every automorphism 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, what generalizes the result from [Bardakov, Nasybullov, and Neshchadim]. Finally, if G is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of G.