  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 $(\mathbb{Z}^{d} \rtimes
P)\times G$.
