The classifying stack of a finite group

Every torsor is finite etale. The given kernel-pair isomorphism trivializes $P\to T$ over the cover $P\to T$. Over an affine open $T_0\subseteq T$, choose finitely many affine opens in $P_{T_0}$ whose open images cover $T_0$, using [F2]. Their disjoint union $V\to T_0$ is an affine faithfully flat quasi-compact fppf refinement. Over $V$ the torsor is $G_V$, hence finite etale. Its canonical finite-etale descent datum is effective by [F1], giving a finite etale $T_0$-scheme $Q$. The local isomorphism $Q_V\cong P_V$ and its inverse descend as morphisms by [F1], and their composites are identities by uniqueness. Thus $P_{T_0}\cong Q$, and these identifications glue over target opens. Every torsor is therefore finite etale and surjective over any $T$, with no Noetherian hypothesis. Base change preserves its torsor kernel-pair isomorphism, so the stated fibred category exists.

$BG$ is a stack in groupoids. For a compatible family of torsors on any fppf covering, work over an affine target open and choose a finite affine refinement of that covering by the open-image argument of [F2]. The underlying finite-etale covers descend effectively by [F1] and step 1.1. Their action maps and the inverse of the torsor kernel-pair isomorphism descend by morphism descent in [F1]; their identities hold because they do after pullback. Surjectivity is detected on the cover, so the descended scheme is a torsor. Equivariant morphisms descend uniquely for the same reason, and uniqueness glues these constructions over target affine opens. This proves both effective object descent and the sheaf condition for morphisms, hence $BG$ is a stack.

Diagonal and presentation. For two torsors $P,Q$ over $T$, the finite etale surjective cover $P{\times }_{T}Q\to T$ trivializes both by step 1.1. The sheaf of equivariant isomorphisms becomes $G$ on this cover, with transitions induced by the two trivializations. Its cocycle is canonical, so [F1] represents this Isom sheaf by a finite etale $T$-scheme; this is precisely the base change of the diagonal of $BG$. The trivial torsor supplies $\mathrm{Spec}k\to BG$. Its base change along a torsor $P\mathrm{/}T$ is $P$ itself: an equivariant map $G_{T^{\mathrm{\prime }}}\to P_{T^{\mathrm{\prime }}}$ is uniquely determined by the image of the identity. Thus the base change is finite etale and surjective, hence smooth and surjective. These representable base changes establish a scheme presentation and representable diagonal, so $BG$ is algebraic by [F3]. This checks arbitrary schemes $T$, including schemes with nontrivial torsors, rather than asserting that all global trivializations exist.

Automorphisms and inertia. For the trivial torsor $G_T$ over $T$, every automorphism of right torsors is left multiplication by an element of $G$: if $\phi \text{\hspace{0.17em}⁣}:G_T\to G_T$ satisfies $\phi (pg)=\phi (p)g$, then $\phi (p)=\phi (1)p$ for a locally constant $\phi (1)\in G(T)$, and conversely every such left multiplication is an automorphism. Hence the automorphism sheaf of the trivial torsor is $G_T$ by left multiplication. If $G$ is abelian, then also every automorphism of an arbitrary torsor $P$ is right translation by a section of $G_T$: an automorphism is $\phi (p)=p\cdot g(p)$ with $g(ph)=h^{-1}g(p)h=g(p)$ by commutativity, so $g$ descends along the fppf map $P\to T$ to a section of $G_T$ by [F1]. Consequently, for abelian $G$ the inertia stack satisfies ${\mathcal{I}}_{BG}\cong G_k{\times }_{k}BG$, and over the trivial torsor in $BG(\mathrm{Spec}k)$ the inertia objects are exactly the pairs $(\text{trivial torsor},g)$ with $g\in G$; for nonabelian $G$ the automorphism sheaf of an arbitrary torsor is only a form of $G$, and the stronger product description is asserted only when $G$ is abelian, as in the trivial-torsor case.