Seems like WA is interpreting incorrectly your input "inverse fft of fft of {0,1,2,3}".
Wofram Alpha interprets it as: Fourier[{Fourier[{0,1,2,3}]}]^-1.
You can get this input by doing in Mathematica == inverse fft of fft of {0,1,2,3}.
which is different from InverseFourier[Fourier[{0, 1, 2, 3}]]
