I'm trying to describe Bravais lattice point groups as permutations of lattice points. In doing so, I encounter a problem: I can only find descriptions of Bravais lattice point groups in terms of crystal systems. However, Bravais lattices can be in the same crystal family and have the "same" point group but have different permutation groups for their lattice points.
To make this clear, we can take an example in 2d. As explained in this Wikipedia page, both the rectangular and centered rectangular lattices are part of the same crystal family and can be seen as having the same point group $D_2$. However, upon application of a symmetry transformation points are not permuted in the same way for both lattices.
Another way to see this is by considering a primitive unit cell for each lattice and look at its symmetry. The rectangular unit cell has symmetry elements : identity, 180 rotation, horizontal reflection, vertical reflection.
Clearly in both cases, applying a symmetry element will give you different permutations of lattice points and that's what I'm interested in. We could see them as different groups, the operations are different and they are different subgroups of the square.
But since they are isomorphic and can be mapped to each other by choosing the centered rectangular unit cell instead of the rectangular one, they are both labeled as $D_2$ on Wikipedia and any resource I could find.
My question : Is there a classification of Bravais lattice point groups in 3D that actually accounts for these differences? Alternatively, is there a description of the shapes of all primitive cells (that doesn't use base, body or face-centered unit cells)? With that, I could find these groups by myself by looking at the symmetry of each shape, although that would be tedious.