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 $G\ltimes X$.

Let $F\colon (A\rightrightarrows X)\to (B\rightrightarrows Y)$ be a morphism of groupoids (a functor).
We say that $F$ is an *equivalence of groupoids* if it is an equivalence of categories.

Let $x\in X$. We denote by $A(x)$ the set of arrows $a\colon x\to x$.
We denote by $\pi_0(X)$ the set of connected components of $X$
(i.e., the set of equivalence classes in $X$ with respect to the equivalence relation induced by $A$).
We say that a morphism $F$ as above is a *weak equivalence of groupoids* (or a *quasi-isomorphism*) if
$\pi_0(F)\colon \pi_0(X)\to \pi_0(Y)$ is a bijection and, for any $x\in X$, the induced homomorphism
$F_x\colon A(x)\to B(y)$ is an isomorphism, where $y=F(x)$.

Question 1.Is it true that any weak equivalence of groupoids is an equivalence?

Now assume that a group $\Gamma$ acts on our groupoid $A\rightrightarrows X$. We say that $A\rightrightarrows X$ is a $\Gamma$-groupoid. My favorite example of a $\Gamma$-groupoid comes from an action of an algebraic group $\mathcal{G}$, defined over a field $k$, on a $k$-variety $\mathcal{X}$. Let $k_s$ denote a separable closure of $k$, then we set $\Gamma:={\rm Gal}(k_s/k)$, and we consider the action groupoid $\mathcal{G}(k_s)\ltimes\mathcal{X}(k_s)$, on which $\Gamma$ acts.

By a *weak equivalence of $\Gamma$-groupoids* we mean a $\Gamma$-functor $F\colon (A\rightrightarrows X)\to (B\rightrightarrows Y)$
that is a weak equivalence of groupoids.
By an *equivalence of $\Gamma$-groupoids* we mean a $\Gamma$-functor $F\colon (A\rightrightarrows X)\to (B\rightrightarrows Y)$
such that there exists a a $\Gamma$-functor $F'$ in the opposite direction and each of the composite functors $F\circ F'$ and $F'\circ F$ is
$\Gamma$-naturally-isomorphic to the corresponding identity functor.

Question 2.Is it true that any weak equivalence of $\Gamma$-groupoids is an equivalence of $\Gamma$-groupoids?

I expect the answer "No" to Question 2, but I cannot construct a counter-example.