I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-sets.

For example Pirashvili prove, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-group (i.e. a group object in the category of $\Gamma$-sets).

What I would like to know if this can be done in a "nice way" similarly to how quasi-category can be obtained from Segal spaces. (for example, Pirashvili's argument is quite indirect and the functoriality property of its construction seems unclear). More precisely:

Consider the Bousfield-Friedlander model structure on $\Gamma$-spaces (here "space", will mean pointed simplicial sets) whose equivalence are the stable equivalence.

If $X$ is a simplicial set I denote by $X_0$ its set of verticies $X([0])$. If $X$ is a $\Gamma$-space, $X_0$ the application of this construction ``levelwise in $\Gamma$''. i.e.:

$$ X_0 : \Gamma^{op} \overset{X}{\rightarrow} sSet_* \overset{(\_)_0}{\rightarrow} Set_* $$

Question: Given $X$ a $\Gamma$-space fibrant in the Bousfield Friedlander model structure, is the natural morphism:

$$ X_0 \rightarrow X$$

a (stable) weak equivalence ?

I believe this the right question to ask, but if there is something very similar with a positive answer (maybe "fibrant in the Bousfield-Friedlander model structure" is not quite the right condition ?), please let me know !