Let $G$ be a finite group and $q:G\rightarrow G:g\mapsto g^2$ the square map. Now, if $A$ is a subset of one fibre of $q$, i. e. $$a^2=b^2$$ holds for all $a,b\in A$, is there always some $g\in G$ such that $$gA=A^{-1}\,\,?$$ (note: $gA:=\{ga|a\in A\}$ is simply the left-translate of the set $A\subseteq G$ with the element $g\in G$.) Calculations with GAP for groups up to order 60 support this conjecture, however I was unable to prove it and actually I'm not convinced that it holds for larger groups. Some related well-known (http://groupprops.subwiki.org/wiki/Square_map) facts are:
The mapping $q$ is bijective iff $2\not|\, |G|$. (So only groups of even order are interesting w.r.t our question.)
The mapping $q$ is a homomorphism iff $G$ is Abelian. (However I'm interested in the non-Abelian case as well.)