Given an even permutation $\sigma,$ what more is required for it to be the square of some other permutation? Is it enough that there be an even number of transpositions in its cycle decomposition?
I looked at the case $\sigma=(12)(3456)$ and could not find a squareroot for that, or others I tried having an odd number of transpositions in their decompositions.
I would be interested in any specific counterexample having an odd number of transpositions, and also whether more is required for a permutation to be a square.