  We introduce a new class of transitive permutation groups which properly
contains the automorphism groups of vertex-transitive graphs and digraphs. We
then give a sufficient condition for a quotient of this family to remain in the
family, showing that relatively straightforward induction arguments may
possibly be used to solve problems in this family, and consequently for
symmetry questions about vertex-transitive digraphs. As an example of this, for
$p$ an odd prime, we use induction to determine the Sylow $p$-subgroups of
transitive groups of degree $p^n$ that contain a regular cyclic subgroup in
this family. This is enough information to determine the automorphism groups of
circulant digraphs of order $p^n$.
