I don't follow your hints: if , say and then there is NO unique solution to the equation in S_3....so why do you hint to take to be any finite group? Unless you meant that ANY finite group will fail the test...but then this hardly answers the OP's question, doesn't it?
I, for one, cannot think of any group at all, abelian or not, finite or not, that the required condition is true...
finding a non-abelian group with that property doesn't seems to be that easy! what is obvious is that such groups would have to be infinite.
to prove that it has the above property see that for any and integer we have so if and are given, then the
only which satisfies the equation is where and