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.
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. I thought I had a way to make $G/[G,G]$ act in a very easy way on conjugacy classes of $G$, but David Speyer drew my attention to the fact that it must be much less obvious. Hence for the moment my question is - is there a way to repair the original question along these lines? For instance, is there an action of $G/[G,G]$ on conjugacy classes of $G$ for which there is 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 $G$?