I think this question is related to this other question, but somewhat different.
Let there be a hidden datum $D$ that we observe using a hash function $H_1$, $h_1 = H_1(D)$. There's another hashed value that we get from $h_2 = H_2(D)$. Is there a way to pick $H_1, H_2$ such that there is another function $G$ that we can apply to get $h_2 = G(h_1)$, without having to know about the true value of $D$? Or is it that the only way to get $h_2$ is by application of $H_2(D)$?
In my real world problem, $H_1$ is already a fixed hash function. But if one were to relax the requirement on $H_1$ so that it was allowed to be any kind of encryption function, would that make the above possible? (Assuming that $H_1$ being a hash would make it impossible.)