*Setup:* Let's work in the category of schemes over $\mathbb C$. Let $G$ be a finite group. Let $BG=[pt/G]$ be the classifying stack of principal $G$ bundles. This is a fiberd category over the big etale site of complex schemes. The groupoid of sections of $BG$ over $U$ is the category of principal $G$ bundles over $U$. I will also appreciate it if someone can answer the question in the topological category.

*Question:* What are the automorphisms of $BG$, as a stack over $\mathbb C$?

*What to Expect:* Since automorphisms of $BG$ are functors, $Aut(BG)$ should be a category. It does not quite make sense to say that two automorphisms are the same. We can only say that two automorphisms are isomorphic. An automorphism of $BG$ might have nontrivial (2-)automorphisms.

*What I've Tried:* I trived the case when $G$ is abelian and got lost even for this simpler case. Imaging $BG$ as a point with automorphisms $G$, I expect that the objects of $Aut(BG)$ are $Aut(G)$, since a functor sends the point to itself and sends automorphisms to automorphisms. But when I look at the groupoid $BG(U)$, I have some difficulty. Suppose $a,b\in BG(U)$ are different objects and $\phi\in G$, there is no canonical way that $\phi$ acts on $Hom(a,b)$. However, although this is a big trouble, it's not hopeless. It seems that we can pick an distinguished element in $Hom(a,b)$ so that we can identify it with $G$. We can pick an isomorphism of a groupoid with its skeleton.

*My Guess:* Let me have a guess. I guess that $Aut(BG)=[Aut(G)/G]$. This is the quotient groupoid given by the conjugation map $G\to Aut(G)$.