Ill send you the slides of my talk I'll give on Wednesday, feel free to make suggestiones.

you can reduce the study of the groups who represent faithfully into O(2) x O(1) to the study of those who represent faithfully into O(2), those are the cyclic groups and the diedral groups. For the cyclic groups it is easy to check that they have no irreducible 3dimensional representations and that their Alt²(E) are all trivial, and for the diedral groups it's probably kinda easy if I bothered to spend time on it

it's also probably in the literature somewhere

and then we can see Alt²(E) is 1 on the rotations and -1 on the reflections, for any 2dim real irrep E