Partially-elementary end extensions of countable models of set theory

Let KP denote Kripke-Platek Set Theory and let M be the weak set theory obtained from ZF by removing the collection scheme, restricting separation to Δ₀-formulae and adding an axiom asserting that every set is contained in a transitive set (TCo). A result due to Kaufmann shows that every countable model, M, of KP+Πₙ-Collection has a proper Σₙ₊₁-elementary end extension. Here we show that there are limits to the amount of the theory of M that can be transferred to the end extensions that are guaranteed by Kaufmann's Theorem. Using admissible covers and the Barwise Compactness Theorem, we show that if M is a countable model KP+Πₙ-Collection+Σₙ₊₁-Foundation and T is a recursive theory that holds in M, then there exists a proper Σₙ-elementary end extension of M that satisfies T. We use this result to show that the theory M+Πₙ-Collection+Πₙ₊₁-Foundation proves Σₙ₊₁-Separation.