Clones Closed under Conjugation I: Clones with Constants

We show that if G ≤ SA is a permutation group on a finite set A satisfying |A| ≥ 3, then the set of G-closed clones on A that contain all constant operations is finite if and only if G = SA, AA, AGL(1, 5) (|A| = 5), PSL(2, 5) (|A| = 6), PGL(2, 5) (|A| = 6), PGL(2, 7) (|A| = 8), PGL(2, 8) (|A| = 9), or PΓL(2, 8) (|A| = 9).