  We show that for any continuous monotonic fixed-point free automorphism $f$
on a $\sigma$-compact subgroup $G\subset \mathbb R$ there exists a binary
operation $+_f$ such that $\langle G, +_f\rangle$ is a topological group
topologically isomorphic to $\langle G, +\rangle$ and $f$ is a shift with
respect to $+_f$. We then show that monotonicity cannot be replaced by the
property of being periodic-point free. We explore a few routes leading to
generalizations and counterexamples.
