Let $A \subseteq B$ be commutative noetherian domains. Of course, if $M$ is an $A$-module, then $M \otimes_A B$ is a $B$-module. I am curious to know if there exist additional conditions on $A$ and $B$, such that every $B$-module $N$ is necessarily of the form $M \otimes_A B$ for some $A$-module $M$.

I do not mind to assume one or more of the following additional conditions: $A$ is a UFD (but I do not want to assume that $B$ is a UFD). $A$ is regular. $B$ is a complete intersection ring (but I do not want to assume that $B$ is regular). $B$ is a faithfully flat $A$-module. $B$ is a free $A$-module.

I once ran into a paper (unfortunately I cannot find it now) which calls such $N$ extendable (maybe that paper answers my curiosity?).