Let G be a finite group of order n2. A perfect square root of G is a subset X of G such that |X| = n and X2 = G. Neither generalized dihedral groups nor groups of nilpotency class two have perfect square roots.
Abhyankar, Kashi and Grossman, Daniel, "Square Roots of Finite Groups" (1994). Mathematical Sciences Technical Reports (MSTR). 123.