Two subgroups $H_1, H_2$ of a group $G$ are conjugate iff $G/H_1$ and $G/H_2$ are isomorphic

If $\phi : G/H_1 \to G/H_2$ is a homomorphism of $G$-sets, and $\phi([1])=[g]$, then it follows more generally that $\phi([x])=[xg]$. But $\phi$ should be well-defined, i.e. $y^{-1} x \in H_1$ implies $(y g)^{-1} x g \in H_2$. This reduces to $g^{-1} H_1 g \subseteq H_2$. Conversely, this relation implies that $\phi([x]):=[xg]$ is a well-defined homomorphism of $G$-sets. Similarly, $\phi$ is injective iff $g H_2 g^{-1} \subseteq H_1$. And $\phi$ is automatically surjective. It follows that $G/H_1 \cong G/H_2$ as $G$-sets iff $H_1$ and $H_2$ are conjugated.

As already pointed out in the comments, it is really important to work in the category of $G$-sets here. But there aren't any alternatives anyway. Of course the category of sets is too weak, and the category of groups doesn't make sense since $H_1,H_2$ aren't assumed to be normal.