Finite compactifications of $\omega^* \setminus \{x\}$

We prove that under [CH], finite compactifications of $\omega^* \setminus \{x\}$ are homeomorphic to $\omega^*$. Moreover, in each case, the remainder consists almost exclusively of $P$-points, apart from possibly one point. Similar results are obtained for other, related classes of spaces, amongst them $S_\kappa$, the $\kappa$-Parovi\v{c}enko space of weight $\kappa$. Also, some parallels are drawn to the Cantor set and the Double Arrow space.