  Motivated by problems involving end extensions of models of set theory, we
develop the rudiments of the power admissible cover construction (over
ill-founded models of set theory), an extension of the machinery of admissible
covers invented by Barwise as a versatile tool for generalizing model-theoretic
results about countable well-founded models of set theory to countable
ill-founded ones. Our development of the power admissible machinery allows us
to obtain new results concerning powerset-preserving end extensions and rank
extensions of countable models of subsystems of $\mathsf{ZFC}$. The canonical
extension $\mathsf{KP}^\mathcal{P}$ of Kripke-Platek set theory $\mathsf{KP}$
plays a key role in our work; one of our results refines a theorem of Rathjen
by showing that $\Sigma_1^\mathcal{P}\text{-}\mathsf{Foundation}$ is provable
in $\mathsf{KP}^\mathcal{P}$ (without invoking the axiom of choice).

Logic