I will admit that I always have trouble with visualising this (I remember from a course on solid state physics that I succeeded best at finding the symmetry axis when I had a small model of the molecule in my hands). But this is my trial at an answer.
I take the yellow arrow as the z-direction.
Fe-I:
4-fold rotation axis in z-direction.
2-fold rotation axes in the other directions.
=> 3 perpendicular 2-or-more-fold rotation axes, which determine the axes of the principle component axis system. For the direction of the 4-fold rotation axis there is axial symmetry around it for the electric field gradient tensor.
Fe-II
4-fold rotation axes in all 3 directions.
=> These axes determine the principle component axis system, and there is axial symmetry about all axes.