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
 http://mathoverflow.net/questions/127729.

Now let $\Gamma$ be a group, and assume that $\Gamma$ acts compatibly on $G$ and on $X$ (see my question
 http://mathoverflow.net/questions/130712
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 http://mathoverflow.net/questions/130712 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".