There are several scattered statements about fixed points and obstructions which I'd very much like to see unified in some framework.

To state them let $G$ be a group acting on a connected (1-truncated) groupoid $X$. Firstly let's choose a point in $X$ and take $X=BAut(\pi_1(X))=:BA$. I'd like to **find a unified homotopy theoretic approach** for the following statements:

 1. There's an obstruction in $H^2(G,A)$ for the existence of homotopy fixed points.

 2. When the obstruction vanishes $\pi_0(X^{hG})\cong H^1(G,A)$


**I'm hoping both of these statements can be explained using arguments about the following fiber sequence (and perhaps some additional close constructions):**

$$BA \to E \to BG$$

Where $E$ is the $A$-gerbe corresponding to the $G$ action on $X$. Said differently we have by assumption a map $G \to Aut(BA)$ which we can $B(-)$ to get $BG \to BAut(BA)$ which we can use to pullback the universal fibration $BA \to BAut_*(BA) \to BAut(BA)$ to get the above fiber sequence.

Ideally both $H^2(G,A)$ and $H^1(G,A)$ will appear in the same long exact sequence of homotopy groups for some fiber sequence.