8.84 (1982)

We say that an automorphism $\varphi$ of the group $G$ is a pseudo-identity if, for all $x \in G$, there exists a finitely generated subgroup $K_x$ of $G$ such that $x \in K_x$ and $\varphi|_{K_x}$ is an automorphism of $K_x$. Let $G$ be generated by subgroups $H, K$ and let $G$ be locally nilpotent. Let $\varphi : G \to G$ be an endomorphism such that $\varphi|_H$ is pseudo-identity of $H$ and $\varphi|_K$ is an automorphism of $K$. Does it follow that $\varphi$ is an automorphism of $G$? It is known that $\varphi$ is a pseudo-identity of $G$ if, additionally, $\varphi|_K$ is pseudo-identity of $K$; it is also known that $\varphi$ is an automorphism if, additionally, $K$ is normal in $G$.