Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a C ∞ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphism group of a transversely holomorphic foliation or a riemannian foliation is a strong ILH Lie goup in the sense of Omori.