Is this true about groups

Question

Let $G$ and $H$ are two non-abelian group. Let [G,G] be the commutator subgroup of $G$ then $G/[G, G]$ will be abelian.

Question : Is it true that $G$ and $H$ are isomorphic iff $G/[G,G]$ and $H/[H, H]$ are isomorphic.

Answer 1

A consequence of that would be that any two groups $G$ such that $G=[G,G]$ are isomorphic. Since that's clearly not the case…

Answer 2

For a perfect group $G$ you have that $G/[G,G]$ is the trivial group. Thus, the statement is false (take for example the alternating groups $A_5$ and $A_6$).