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We study orbit codes in the field extension   \begin{document}$  \mathbb{F}_{q^n} $\end{document}  . First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of   \begin{document}$  \mathbb{F}_{q^n} $\end{document}  . We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.

Automorphism groups and isometries for cyclic orbit codes