Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then ?G,a??G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates ag=g?1ag of a by elements of G generate a semigroup denoted by ?ag|g?G?. We classify the finite permutation groups G on a finite set X such that the semigroups ?G,a?, ?G,a??G, and ?ag|g?G? are regular for all transformations of X. We also classify the permutation groups G on a finite set X such that the semigroups ?G,a??G and ?ag|g?G? are generated by their idempotents for all non-invertible transformations of X.