Admissible reversing and extended symmetries for bijective substitutions

In this paper, we deal with reversing and extended symmetries of shifts generated by bijective substitutions. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how to check them. Moreover, we show that, for any finite group $G$ and any subgroup $P$ of the $d$-dimensional hyperoctahedral group, there is a bijective substitution which generates an aperiodic hull with symmetry group $mathbb{Z}^{d}times G$ and extended symmetry group $(mathbb{Z}^{d} rtimes P)times G$.