Spin connection for a paralellization takes more general forms than $SO(3,1)$ in different spacetime topologies?

I’m interested in a frame bundle over spacetimes with different topologies. In the trivial case of Minkowskian space ($mathbb{R}^{3,1}$), a frame (or tangent space) at one point is going to be related to a frame (or tangent space) at another point via an $SO(3,1)$ rotation. Hence we get a spin connection valued in that group. In fact we can always reduce the structure group locally to this for a Lorentzian manifold (ie spacetime). I’m more interested in global structures on spacetimes.

Let’s consider a spatial slice in a closed Friedman-Lemaitre-Robertson-Walker type spacetime, which has a spatial $S^{3}$ topology. A frame (or the tangent space) at an arbitrary point on the space is related to a frame (or the tangent space) at another point via an $SO(4)$ rotation, and any spatial triad (or tangent spaces) could be thus related.

Considering a full spacetime then with topology $S^{3}timesmathbb{R}$ , wouldn’t the structure group for a parallelization necessarily be $SO(4,1)$? In other words wouldn’t the covariant derivative in an oriented orthonormal frame (the parallelization) necessarily have an $SO(4,1)$ spin connection? Which itself would reduce to the standard $SO(3,1)$ spin connection in the neighborhood of a point?
If you want to formulate your answer in terms of fiber bundles that’s totally fine! (: