as far as i know, one of the key ideas of general relativity is that of general covariance, which states that the laws of physics are invariant under general coordinate transformations (not only inertial). this idea seems to be realized mathematically as diffeomorphism invariance, a symmetry under the diffeomorphism group which renders the laws of general relativity independent of the space in question.
this would suggest that mathematically, one seeks to employ the diffeomorphism group $ \operatorname{Diff}(M) $ as the structure group of the theory. however in all of the literature i have come across this is not the case as they tend to refer to the structure group of general covariance as $ GL(k,\mathbb{R})$ where $k$ is the spacetime dimension.
Concretely, we use $ GL(k,\mathbb{R}) $ to build the tangent frame bundle but in this construction, the diffeomorphism group never comes up, why is it that?
If the diffeomorphism group is the "ultimate" group of GR, why use $ GL(k,\mathbb{R}) $? Where does the diffeomorphism group come out?
Concretely, i wonder if $ \operatorname{Diff}(M) $ gets reduced to $ GL(k,\mathbb{R}) $ in some way?
