Let $G$ be a finite group, and let $p:E\to B$ be a $G$-fibration with fibre $F$. The correct framework for studying equivariant sections of $p$ is Bredon cohomology. With the right definitions, most of the classical obstruction theory generalizes to this setting (even for $G$ compact Lie - details appear in a paper of Mukherjee-Mukherjee, for example).

One could also apply the Borel construction to $p$ to obtain a map $$1\times_Gp:EG\times_G E\to EG\times_G B,$$ where $EG\to BG$ is a numerable classifying bundle. (That this is a fibration was shown in the answers to the question Does the Borel functor take equivariant fibrations to fibrations?) The fibre is again $F$, and the obstructions live in the Borel equivariant cohomology groups $H^{n+1}(EG\times_GB;\pi_n(F))=:H^{n+1}_G(B;\pi_n(F))$. Clearly, equivariant sections of $p$ give sections of $1\times_G p$, but the converse is less clear (and probably false).

My questions is about whether anyone has studied the relationship between the obstruction theory of $1\times_G p$ and that of $p$? Perhaps in terms of the spectral sequence $$H^p(G;H^q(B;\pi_*(F)))\Rightarrow H^*_G(B;\pi_*(F))?$$

Any insight or references would be appreciated!