Problem:
An octahedron is inscribed in a cube whose edge length is 2. A cube is inscribed inside the octahedron. All the 4-fold, 3-fold and 2-fold symmetry axes of these objects coincide.
Method:
I calculated the edge length of the octahedron first. Imagine the 2D slice to look like this, then used the Pythagorean Theorem to find that the octahedron edge length was $\sqrt{2}$. All edges are equal in length for the octahedron, so I proceeded to calculate the distance from the center of any triangular face to the middle of any edge on the same face ($\frac{1}{\sqrt{6}}$). Using two different lines from center to edge on the same triangular face, I used the Pythagorean Theorem to solve for the inner cube edge ($\frac{1}{\sqrt{3}}$).
Questions:
Is my solution correct? I appreciate any suggestions or alternate solutions.
Reference Images:
No, it is not correct.
The calculation of the octahedron sidelength is good, but you made a false assumption in calculating the sidelength of the inner square. I could tell you what it is, but I suspect you may already know, or at least where to look. So I'll let you think about it first. Leave a comment if you can't find it.
As for an alternative approach, try solving it analytically in coordinates (hint: the center of a triangle is the average of its 3 corners). Even if you want a geometric proof, the analytic calculation will be quite instructive.