It is well known that General Relativity (GR) has a symmetry under diffeomorphisms $\text{Diff}(M)$ (see for example Wald's book on GR), that is, if the metric $g$ is a solution of the Einstein's field equations, so is the metric $\phi^*g$ for a diffeomorphism $\phi:M\to M$.
On the other hand, it's also claimed that GR can be viewed as a gauge theory (in the more general sense, not strictly in the sense of particle physics) with the structure group $GL(4,\mathbb{R})$ since the bundle in question would be the usual frame bundle. Is this a contradiction?
While I understand those diffeomorphims can be seen as "active" transformation on the coordinates, the group $\text{Diff}(M)$ is fundamentally different from $GL(4,\mathbb{R})$ and, in fact, the former is infinite dimensional. So what is the true structure/gauge group of GR?