This question is motivated by this one. Basically, the remark of Bugs Bunny in comments made me think - OK, there is an ambiguity in choosing between a Young diagram and its transpose when constructing irreps of $S_n$, but it is somewhat remarkable: it corresponds in terms of irreps to tensoring representations with the sign representation. There is also of course the remark of Gjergji Zaimi suggesting that for abelian groups the ambiguity is maximal possible, since we are literally dealing with a group vs its dual. This of course makes one wonder if the group of characters $G/[G,G]$ is somehow responsible for the key ambiguities, hence this (possibly very naive) question.

Clearly, $$hcgc^{-1}=h(cgc^{-1}g^{-1})g$$ for all $c,g,h\in G$. Also, $[G,G]$ is a normal subgroup, so for every $z\in [G,G]$ and every $h$ in $G$ we have $hz=uh$ for some $u\in [G,G]$, so $$hzg=uhg=u(hgu)u^{-1}=u(hgu)u^{-1}=u(hu(u^{-1}gu))u^{-1}.$$ These two equations seem to suggest that the obvious left action of $G$ on itself gives rise to the action of $G/[G,G]$ on the set of conjugacy classes of $G$. Also, of course the dual group of $G/[G,G]$ acts on irreps of $G$: that dual group is just the group of one-dimensional irreps of $G$, and we can tensor irreps of $G$ with those. Now, a vague question is - is this somehow a way to repair the original question? For instance, is there a chance to have a natural (in some sense) bijection between the orbits of $G/[G,G]\simeq (G/[G,G])^\vee$ on the conjugacy classes of $G$ and the irreps of $G$?