Suppose $j : M \to N$ is an elementary embeddings between transitive models of ZFC. Everyone knows that if $G$ is $\mathbb{P}$-generic over $M$, $H$ is $j(\mathbb{P})$-generic over $N$, and $j[G] \subseteq H$, then the map can be extended to $\hat{j} : M[G] \to N[H]$.

Suppose now that $\mathbb{P}$ is a complete boolean algebra generated by some name $\tau$ for a subset of an ordinal $\kappa$. Suppose $A \subseteq \kappa$ is generic. In $M[A]$, we compute a generic for $\mathbb{P}$ corresponding to $A$. Now assume $B \subseteq j(\kappa)$ is $j(\mathbb{P})$-generic over $N$, and for all $\alpha < \kappa$, $\alpha \in A$ iff $j(\alpha) \in B$. Is it always true that we can extend $j$ to $\hat{j} : M[A] \to N[B]$? In other words, is it necessarily the case that the generic filter $G$ computed from $A$ has the property that $j[G] \subseteq H$, where $H$ is the generic filter computed from $B$?