  We study orbit codes in the field extension ${\mathbb F}_{q^n}$. 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 ${\mathbb F}_{q^n}$. 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.
