Let $\mathcal X$ be a presentable $\infty$-category. Then the stabilization $Stab(\mathcal X)$ of $\mathcal X$ is the universal presentable stable category on $\mathcal X$.
Conversely, if $\mathcal A$ is a presentable stable $\infty$-category, then we can ask which presentable $\infty$-categories $\mathcal X$ have $Stab(\mathcal X) \simeq \mathcal A$. There's always at least one such $\mathcal X$, namely $\mathcal A$ itself. In particular, I would like to know an answer to the following:
Question 1: Let $\mathcal A$ be a presentable stable $\infty$-category. Under what conditions does there exist an $\infty$-topos $\mathcal X$ such that $Stab(\mathcal X) \simeq \mathcal A$?
For a closely related question, let $StPr^L$ denote the $\infty$-category of presentable stable $\infty$-categories and left adjoint functors. Let $Logoi$ denote the $\infty$-category of $\infty$-topoi, with geometric morphisms pointing in the direction of their inverse images.
Question 2: Does the functor $Stab : Logoi \to StPr^L$ have a left or right adjoint?
If the answer to Question 2 is affirmative, then one might approach Question 1 by asking for criteria ensuring that the unit / counit of the adjunction is an equivalence. Alternatively, one might wonder
Question 3: Note that the functor $Stab : Pr^L \to StPr^L$ has both a left adjoint $L$ and a right adjoint $R$. For which presentable stable $\infty$-categories $\mathcal A$ is $L\mathcal A$ or $R \mathcal A$ an $\infty$-topos?
Question 4: For example, let $G$ be a compact (even finite, say) Lie group. Is the category $Spt_G$ of genuine $G$-spectra the stabilization of an $\infty$-topos?