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

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.