Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

Let a be a non-invertible transformation of a nite set and let G be
a group of permutations on that same set. Then h G; a i nG 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 g 2 G generate
a semigroup denoted hag j g 2 Gi. We classify the nite permutation groups G
on a nite set X such that the semigroups hG; ai, hG; ainG, and hag j g 2 Gi are
regular for all transformations of X. We also classify the permutation groups
G on a nite set X such that the semigroups h G; a i nG and h ag j g 2 Gi are
generated by their idempotents for all non-invertible transformations of X.