  A hyperbolicity cone is said to be rank-one generated (ROG) if all its
extreme rays have rank one, where the rank is computed with respect to the
underlying hyperbolic polynomial. This is a natural class of hyperbolicity
cones which are strictly more general than the ROG spectrahedral cones. In this
work, we present a study of the automorphisms of ROG hyperbolicity cones and
their derivative relaxations. One of our main results states that the
automorphisms of the derivative relaxations are exactly the automorphisms of
the original cone fixing a certain direction. As an application, we completely
determine the automorphisms of the derivative relaxations of the nonnegative
orthant and of the cone of positive semidefinite matrices. More generally, we
also prove relations between the automorphisms of a spectral cone and the
underlying permutation-invariant set, which might be of independent interest.
