Let n be given. Can you find nxn invertible matrices T and S over Z2 such that T2S = ST? If so, can you furthermore have both S and T be 'primitive' in the sense that neither Sk nor Tk are the identity for any positive k < 2n-1?
Is this possible for some n, for no n, for most n, ...?
An example of the first condition being satisfied in a continuous setting --- affine transformations over real vectors --- is T being a translation and S being a scaling by a factor of two: translate and scale up is the same thing as scale up and translate twice as much.
The second condition is just asking that S and T considered separately are good LFR pseudorandom number generators.
Hmm, my gut instinct after thinking about it for five minutes is: probably true for hardly any n, if any at all.
I'm demanding that all the 2n-1 powers of T are distinct, and that (T, T^2, T^4, T^8,...) are all in the same conjugacy class, and also that (T^3, T^6, T^12, T^24, ...) are all in the same conjugacy class, and also (T^5, T^10, T^20, T^40, ...), and so on starting at every odd power of T. So I'm making really strong constraints on the structure of the conjugacy classes of GF(Z2), and I bet reality doesn't conform to them.