Endomorphisms with pseudoidentical restrictions (Q1905326)

This note answers question 8.84 of P. Hilton from the Kourovka Notebook. Let \(G\) be a locally nilpotent group generated by its subgroups \(H\) and \(K\), \(\varphi\) be an endomorphism of \(G\) where \(\varphi|_K\in\text{Aut }K\) and \(\varphi|_H\) is a pseudoidentical automorphism of \(H\), i.e., for each \(x\in H\) there exists a finitely generated subgroup \(H_x\) of \(H\) for which \(\varphi|_{H_x}\in\text{Aut }H_x\). The question is whether, under these conditions, \(\varphi\in\text{Aut }G\). As noted in the Kourovka Notebook, the answer is yes if the automorphism \(\varphi|_K\) is pseudoidentical or \(K\) is normal in \(G\). Nevertheless, the answer to Hilton's question is no in general. Theorem. There exists a metabelian group \(G\) generated by its subgroups \(H\) and \(K\) and a noninvertible endomorphism \(\varphi\in\text{End }G\) such that \(\varphi|_K\in\text{Aut }K\) and \(\varphi|_H\) is the identity automorphism of \(H\).