That is the traditional Fourier transform, except it can be a cyclic group of any size, doesn't need to be a power of 2. (Though FFTs with 2^n input size are particularly easy to implement.)

And it's not permutation invariant.

I was careless in my thinking, thanks for the correction. I was imagining since you sum the group elements in any order, there is a permutation invariance. But the group elements themselves play the role of the "token index" and group elements are not interchangeable. To actually make this idea interesting, one would have to use a group in which the choice of group element assigned to each input token would not affect the result.

You mean for the group operation to be standard modular addition? In that case (as a sibling comment says) you'll recover the classic discrete Fourier transform.