On a question of B. H. Neumann

The automorphism group of a free group $\mathrm{Aut}(F_k)$ acts on the set of generating $k$-tuples $(g_1,\dots,g_k)$ of a group $G$. Higman showed that when $k=2$, the union of conjugacy classes of the commutators $[g_1,g_2]$ and $[g_2,g_1]$ is an orbit invariant. We give a negative answer to a question of B.H. Neumann, as to whether there is a generalization of Higman's result for $k \ge 3$.