A note on purely infinite corona algebras and extensions

Let $\mathcal{A}$ be a separable nuclear C*-algebra, and $\mathcal{B}$ be a nonunital separable simple $\mathcal{Z}$-stable C*-algebra. Continuing the work from Gabe-Lin-Ng, we classify all essential extensions, with large complement, of the form $$0 \rightarrow \mathcal{B} \rightarrow \mathcal{E} \rightarrow \mathcal{A} \rightarrow 0,$$ for the following cases: i. $\mathcal{C}(\mathcal{B})$ is properly infinite, and the extension is full. ii. $\mathcal{C}(\mathcal{B})$ is purely infinite (though not necessarily simple). We also have some more general results.