Remark
The notation Gx does not explicitly use f.
But the stabilizer does depend on it.
For instance, if G = S4
and x = (1,2), then
- Gx = {e} if f = L,
left multiplication (or R, right inverse multiplication);
- Gx = {e, (1,2), (3,4), (1,2)(3,4)} if f = C,
conjugation.