(I find it easier to think in terms of Grothendieck pretopologies, instead of topologies. If this annoys any experts, please forgive me.)

Suppose that $C$ is a (full) subcategory of a category $D$. Furthermore, let $J$ be a Grothendieck pretopology on $C$ and let $K$ be a pretopology on $D$ (i.e., for an object $X$ of $C$, $J(X)$ is the collection of covering families for $X$, and similarly for $K$). I am interested in the following situation:

For any object $X$ of $C$, every covering family $\{U_i \to X\}$ for $J$ is also a covering family for $K$ (i.e., $J(X) \subseteq K(X)$).

Is there much precedent for this situation? Am I lucky enough that it can be named using standard terminology?

(Please notice that I am not assuming that the inclusion functor $C \to D$ preserves pullbacks. I realize that this is may create many complications when considering the two topologies. In particular, I don't think that it makes sense to say that the topology generated by $K$ "restricts" to a topology on $C$. This suggests that the answer to my question may not be as simple as saying that the topology from $J$ is subordinate to some restricted topology from $D$, to use terminology from Vistoli's notes, Definition 2.47.)