I am a bit playing around with the octahedral symmetry, and inspired by another exercise I did on finding configurations of the cube such that the stabilizer is equal to a certain group, I tried doing the same for an octahedron. I considered two ways to 'colour' the octahedron. The first is using colouring each face either black or white (the colours do not have to be used equally often), the second is to draw on each face an arrow pointing towards one of the three adjacent vertices.
EDIT 2: My work on the colouring of the octahedron with stabilizer $S_3$ was incorrect, as pointed out by Clément Guérin. Therefore I deleted this. I also just thought about the following. If we allow all rotations that leave one middle-of-two-opposite-faces-connecting line fixed (i.e. stabilizer is $S_3$), this means that we might colour one pair of opposite faces black, and the rest of the $6$ faces white, to get that $S_3$ is the stabilizer of this colouring. To me, this seems correct, and maybe someone can verify this. If so, we found colourings with stabilizer $S_3$, $V_4$, $A_4$ and $C_4$. Similarly I conjecture that a colouring of an octahedron with all but two faces white, and the two black faces are on the bottom of the octahedron, and not adjacent, would have stabilizer $C_2$.
If this is correct, the question is completely answered for the case of a colouring. Perhaps someone also knows how to deal with arrow configurations then, and I will also think about this.
EDIT As Steven Stadnicki pointed out a 'checkerboard pattern colouring (i.e. colour one face black and each adjacent face with a the other colour then the neighbours) the stabilizer is $A_4$ indeed! I coloured the printout below.
Question: Am I right in my claims? $A_4$ was solved by Steven Stadnicki, $V_4$ and $C_4$ by Clément Guérin, and following the thoughts of Clément Guérin I have a conjecture on $C_2$ and $S_3$. The arrow configurations are still unclear to me.
If you find a configuration such that the stabilizer is isomorphic to either $A_4$, $V_4$ or $S_3$, I would find it most helpful if you draw it explicitly on a printout or an octahedron.
I hope I made my question clear, because I am really in doubt about my work and my spatial visualization ability is not that strong.