  A result of Kaufmann shows that if $L_\alpha$ is countable, admissible and
satisfies $\Pi_n\textsf{-Collection}$, then $\langle L_\alpha, \in \rangle$ has
a proper $\Sigma_{n+1}$-elementary end extension. This paper investigates to
what extent the theory that holds in $\langle L_\alpha, \in \rangle$ can be
transferred to the partially-elementary end extensions guaranteed by Kaufmann's
result. We show that there are $L_\alpha$ satisfying full separation, powerset
and $\Pi_n\textsf{-Collection}$ that have no proper $\Sigma_{n+1}$-elementary
end extension satisfying either $\Pi_{n}\textsf{-Collection}$ or
$\Pi_{n+3}\textsf{-Foundation}$. In contrast, we show that if $A$ is a
countable admissible set that satisfies $\Pi_n\textsf{-Collection}$ and $T$ is
a recursively enumerable theory that holds in $\langle A, \in \rangle$, then
$\langle A, \in \rangle$ has a proper $\Sigma_n$-elementary end extension that
satisfies $T$.
