Let $A\rightrightarrows X$ be a groupoid, where $X$ is the set of objects and $A$ is the set of arrows. My favorite example of a groupoid is an action groupoid. If a group $G$ acts on the left on a set $X$, we set $$ A=\{(x,g,y)\mid x,y\in X, g\in G, y=g*x\}, $$ then $A\rightrightarrows X$ with the evident maps is called the action groupoid corresponding to the action of $G$ on $X$. It is often denoted by $G\ltimes X$. If $G$ acts on $X$ transitively, then $G\ltimes X$ is a connected groupoid. Conversely, any connected groupoid is isomorphic to an action groupoid, see the answers to my question Connected groupoids and action groupoids.

Now let $\Gamma$ be a group, and assume that $\Gamma$ acts compatibly on $G$ and on $X$ (see my question Equivalence and weak equivalence of groupoids for a natural example of such action). We say that the $\Gamma$-groupoid $G\ltimes X$ is an action $\Gamma$-groupoid.

Question 1. Is it true that any connected $\Gamma$-groupoid is isomorphic to an action $\Gamma$-groupoid?

Question 2. Is it true that any connected $\Gamma$-groupoid is weakly $\Gamma$-equivalent to an action $\Gamma$-groupoid?

See Equivalence and weak equivalence of groupoids for the definition of a quasi-isomorphism (weak equivalence). We say that two $\Gamma$-groupoids are weakly $\Gamma$-equivalent if they can be connected by a chain of quasi-isomorphisms of $\Gamma$-groupoids.

I expect the answer "No" to Question 1, and therefore I ask Question 2, to which I expect the answer "Yes".

Let $A\rightrightarrows X$ be a groupoid, where $X$ is the set of objects and $A$ is the set of arrows. My favorite example of a groupoid is an action groupoid. If a group $G$ acts on the left on a set $X$, we set $$ A=\{(x,g,y)\mid x,y\in X, g\in G, y=g*x\}, $$ then $A\rightrightarrows X$ with the evident maps is called the action groupoid corresponding to the action of $G$ on $X$. It is often denoted by $G\ltimes X$. If $G$ acts on $X$ transitively, then $G\ltimes X$ is a connected groupoid. Conversely, any connected groupoid is isomorphic to an action groupoid, see the answers to my question Connected groupoids and action groupoids.

Now let $\Gamma$ be a group, and assume that $\Gamma$ acts compatibly on $G$ and on $X$ (see my question Equivalence and weak equivalence of groupoids for a natural example of such action). We say that the $\Gamma$-groupoid $G\ltimes X$ is an action $\Gamma$-groupoid.

Question 1. Is it true that any connected $\Gamma$-groupoid is isomorphic to an action $\Gamma$-groupoid?

Question 2. Is it true that any connected $\Gamma$-groupoid is weakly $\Gamma$-equivalent to an action $\Gamma$-groupoid?

See Equivalence and weak equivalence of groupoids for the definition of a quasi-isomorphism (weak equivalence). We say that two $\Gamma$-groupoids are weakly $\Gamma$-equivalent if they can be connected by a chain of quasi-isomorphisms of $\Gamma$-groupoids.

I expect the answer "No" to Question 1, and therefore I ask Question 2, to which I expect the answer "Yes".