Isomorphic quotient groups

Question

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?

Answer 1

Let $G=\mathbb Z_2\times\mathbb Z_4$. Find two subgroups in $G$ isomorphic to $\mathbb Z_2$ and intersecting trivially such that the quotients of $G$ by them are not isomorphic.

Answer 2

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.