Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable under pullback (i.e. pullbacks of them exist, and they are stable). Each map is to be considered a covering family with a single element.
Consider the following two conditions:
$J$ is saturated: If $U \to V\to X$ is in $J$ then $V \to X$ is in $J$.
$J$ is admissible: $J$ contains the split epimorphisms, and if $U \to V\to X$ is in $J$ and $U \to V$ is a split epimorphism, then $V\to X$ is in $J$.
Now clearly saturated singleton pretopologies are admissible (notice saturated implies $J$ contains the split epis). An example of a saturated pretopology is the class ($K$-epi) of $K$-epimorphisms in a fintely complete category: maps $p:Q\to X$ such that there is a $K$-cover $k:U\to X$ and a map $s:U\to Q$ with $p\circ s = k$ (or more generally local sections for any pretopology, not necc. singleton). Now my question is this:
What is an example of an admissible singleton pretopology which is not of the form ($K$-epi) for some other pretopology $K$?
I know the general situation in categories like $Top$, $Diff$, but not in algebraic settings. As far as I know, pretopologies like ($K$-epi) don't turn up much (I may be wrong, and happy to be corrected), but do admissible singleton pretopologies otherwise arise in algebraic geometry?