No group has commutator isomorphic to S_4
Solution 1:
I see that you want to know how to deal with this kind of problems . How about splitting your question into two paths? if S4 is normal then you know what to do , if not then you have to use this fact "S4 is not normal in A(n)" in order to complete your answer .
As you know the answer to the question already, you can't say that "an argument along these lines is bound to fail"