TITLE: Isomorphic quotient groups
QUESTION [32 upvotes]: Suppose that $G$ is a finite group, and that $H$ and $K$ are normal subgroups of $G$ with trivial intersection, and suppose that $H$ and $K$ are isomorphic. Is it true that the quotient groups $G/H$ and $G/K$ are isomorphic?

REPLY [28 votes]: As Mariano has shown, the answer is a clear no.
The best repair I can think of is the following: suppose that $H$ and $K$ are normal subgroups of a group $G$ such that there exists an automorphism $\varphi: G \rightarrow G$ with $\varphi(H) = K$.  Then $G/H \cong G/K$: indeed, the isomorphism is induced by $\varphi$.  In particular, the above condition holds if $H$ and $K$ are conjugate subgroups of $G$, which is useful enough to be worth remembering.
Note finally that the condition that $H \cap K = \{e\}$ seems to have nothing to do with anything: it neither helps nor hurts the desired conclusion, so far as I can see.