Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$?

Let $$G = \mathbb Z/4\mathbb Z\times\mathbb Z/2\mathbb Z$$ and consider the subgroups $$H = \mathbb Z/4\mathbb Z\times \{e\}\\K=\mathbb Z/2\mathbb Z\times\mathbb Z/2\mathbb Z$$

Then $$G/K\cong G/H\cong\mathbb Z/2\mathbb Z$$ but $H\not\cong K$.

Take, for instance, $G=\mathbb Z/4\mathbb Z \times \mathbb Z/2\mathbb Z$, $H=\mathbb Z/4 \mathbb Z \times \mathbf 0$, and $K=\mathbb Z /2\mathbb Z \times \mathbb Z/2 \mathbb Z$, so that $G/K\cong G/H \cong \mathbb Z/2 \mathbb Z$.

This is (also) not true. Let $G = \mathbf Z^{(\mathbf N)}$, $H = (0)$, and $K = \mathbf Z\cdot e_1$ (where $e_1 = (1, 0,\ldots)$). Then $G/K \cong G \cong G/(0)$, but $K \ne 0 = H$.