Inducing an equivalence of $G$-equivariant categories

Question

Suppose we have an equivalence of triangulated categories $\Phi : \mathcal{A} \to \mathcal{B}$. Let $G$ be a finite group. Are there any methods/conditions for specifying when one has an induced equivalence $\Phi^G : \mathcal{A}^G \to \mathcal{B}^G$ of the associated $G$-equivariant categories?

The particular case I'm interested in is when $\mathcal{A}$ is a semiorthogonal component of $D^b(X)$ and $\mathcal{B}$ is a semiorthogonal component of $D^b(X')$, where $X$ and $X'$ are smooth projective complex varieties. The group $G$ in my case can be something simple like $\mathbb{Z}/2$.

Answer 1

You need some assumptions. But in the case of admissible subcategories, if their equivalence is given by a Fourier-Mukai functor $\Phi_K$ and the object $K \in D^b(X \times X')$ is $G$-equivariant, then $\Phi^G$ can be defined.

See arXiv:1403.7027, Theorem 6.9 for a general statement of this sort.