Let us say a cardinal $\kappa$ *end-extending* if there is a function $F : V_\kappa^{<\omega} \to V_\kappa$ such that:
(a) If $M \subseteq V_\kappa$ is closed under $F$, then $M \prec V_\kappa$.
(b) If $M$ is closed under $F$ and of size $<\kappa$, then there is $N \supseteq M$ closed under $F$ such that $N \cap \sup(M \cap \kappa) = M \cap \kappa$ and $N \cap \kappa \not= M \cap \kappa$.

It is a standard argument to show that measurable cardinals are end-extending, and this reflects below them. It is fairly easy to see that end-extending cardinals must be regular and cannot be successor cardinals (except for the trivial case $\kappa = \omega_1$).

(1) Is being an end-extending limit cardinal equivalent to a well-known large cardinal notion?

(2) Must end-extending limit cardinals be strongly inaccessible?

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