TITLE:  Are there simple conditions on a category C which guaranty that Ind(C) is a Grothendieck topos?
QUESTION [12 upvotes]: The category of finite sets is not a Grothendieck topos, but its Ind category
Ind(Finite-Sets) = Sets
is a Grothendieck topos. Similarly, given a pro-finite group G, the Grothendieck topos of discrete G-sets is the Ind of the category of finite G-sets.
Question: Are there simple conditions on a category C which guaranty that Ind(C) is a Grothendieck topos?
The examples which interests me is of the following kind: I have an infinite sequence of finite groups $(G_i)$ and a corresponding sequence of quotient groups $G_i \to H_i$. I consider the category of sequences $(X_i)$ of finite sets such that each $X_i$ carries a $G_i$-action which factors through $H_i$ for almost all $i$. Even for extreme cases (i.e. $H_i=G_i$, $H_i=1$, or even $G_i=H_1=1$) I don't know if the corresponding Ind-category is a topos. Another possible variant  is to require a uniform bound on the size of the $X_i$'s.

REPLY [7 votes]: In Di Liberti–Ramos González's Gabriel-Ulmer duality for topoi and its relation with site presentations, they state (Theorem 3.17) that $\mathbf{Ind}(\mathscr C)$ is a Grothendieck topos if and only if $\mathscr C$ is extensive and pro-exact (Definition 3.14 therein).
They attribute this result to Carboni–Pedicchio–Rosický's Syntactic characterizations of various classes of locally presentable categories, though it does not appear as an explicit theorem there, so it not such a convenient reference.