Let G and H be finite groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. I am looking for an example where G' is isomorphic to H' and G/G' is isomorphic to H/H' but G and H are not isomorphic.

take $G=S_n$ and $H=A_n\times Z/2Z$ (here $S_n$ is the symmetric group and $A_n$ is the alternating group). 

