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 $X=BAut(\pi_1(X))=:BA$. For the purpose of this question 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.

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 $X=BAut(\pi_1(X))=:BA$. For the purpose of this question 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 total space of 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 (/spectral sequence) of homotopy groups for some fiber sequence.

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.

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 $X=BAut(\pi_1(X))=:BA$. For the purpose of this question 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.