(This post is cross-post in MathOverflow https://mathoverflow.net/questions/381731/symmetry-group-for-noncommutative-manifold-from-spectral-triple)
Is there any notion of symmetry group for noncommutative manifold(in the sense of spectral triple)? Even in the commutative case, given a spectral triple $(A,H,D)$, can we reconstruct the isometry group of the corresponding Riemannian manifold(using Connes's reconstruction theorem)?