at.algebraic topology – A question on recognition of equivariant loop spaces

I have a question about equivariant loop space that has been bothering me, and that I have not been able to find an answer to in the obvious places.

We know from the work of Segal that to give a loop space structure on $X$ is equivalent to producing a simplicial space $X_bullet$ in which $X_0$ is weakly contractible, $X_1$ is weakly equivalent to $X$ and the map $X_n to (X_1)^n$, corresponding to the order-preserving inclusion $(1) to (n)$ taking $0$ to $0$, is a weak equivalence. (see Proposition 1.5 of the article CATEGORIES AND COHOMOLOGY THEORIES by G. Segal)

For the $n$-fold loop spaces case one may see the work of Peter Cobb. The approach is in the same spirit as G. Segal’s investigating of the infinite loop spaces via special $Gamma$-spaces.

Can we have a (similar) description for the equivariant loop space $Omega^V X$ where $X$ is $G$-space and $V$ is a $G$-representation?

Thank you so much in advance. Any help will be appreciated.