Let $G$ be a finite group acting on an abelian category $\mathcal{A}$ in the sense of Deligne (see e.g. Definition 3.1 in Elagin, On equivariant triangulated Categories). Then we can define the equivariant category $\mathcal{A}_G$. The objects are pairs $(E,\{\lambda_g\}_{g\in G})$ where $E\in\mathcal{A}$ is a G-invariant objects of $\mathcal{A}$ and each $\lambda_g\colon E\rightarrow g^\ast E$ is a choice of isomorphism (and morphisms have some compatibility with the group action).
There is also an induced action of $G$ on the bounded derived category, $D^b(\mathcal{A})$, so one can form the equivariant category of that, $D^b(\mathcal{A})_G$.
Q: When is $D^b(\mathcal{A})_G\cong D^b(\mathcal{A}_G)$?
This is true when $\mathcal{A}=Coh(X)$, where $X$ is a scheme and $G$ is a linearly reductive (not necc. finite!) group acting on $X$ (which induces an action on $Coh(X)$ by pullback), see Theorem 9.6 in Elagin, Cohomological descent theory for a morphism of stacks and for equivariant derived categories. I haven't been able to find any references for this for other abelian categories.
This is always true - it's a little deeper in Elagin's paper on equivariant triangulated categories (Theorem 7.1). I don't know how I missed it before!