Quotient of a quotient stack: interesting examples? Let $X$ be a scheme acted on by an algebraic group $G$. Also, let $H$ be an algebraic group acting on the quotient stack $X/G$, for the definition of "act", see Romagny - Group Actions on Stacks and Applications.
I have for a long time been extremely confused about the 2-stack $(X/G)/H$, e.g. when it is a quotient stack, and if it isn't what it can look like. I think to not get bogged down I'll ask my question in a simple case.
Question: Take $X=V$ a vector space acted on linearly by $G\to\operatorname{GL}(V)$, and $H$ an algebraic group.

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*What are examples of $H$ acting on $V/G$ (1a. acting weakly but not strongly) (1b. acting strongly)?

*What is $(V/G)/H$ in these examples?

*In general, when is this double quotient a quotient stack, i.e. of the form $V/G'$?

 A: [Edit: removed some nonsense in first version where I conflated $J$ and $BJ$.]
There's an important example in representation theory of something the situation you're asking about, i.e., a quotient stack $V/G$ with a symmetry group[-scheme] $J$ over $V$ acting inertially (i.e., mapping equivariantly to the family of stabilizer groups, or to the inertia group-scheme of the quotient stack), so that automorphisms of points in $V$ themselves have automorphisms. This comes (by ``looping") from an action of a 2-group (or group-stack) $BJ$ on the stack $V/G$, and the quotient is an honest 2-stack.
This example is due to Ngô as part of his proof of the Fundamental Lemma https://arxiv.org/abs/0801.0446 (the 2-stacky formulation I learned from a lecture of Drinfeld, http://math.uchicago.edu/~drinfeld/langlands/Regular_centralizers.pdf - which says it more clearly than I could).
In this example $V=\mathfrak g$ is a reductive Lie algebra (eg $\mathfrak sl_2$), $G$ is a corresponding group (e.g. $SL_2$), but $H=J$ is not quite a group, it's a family of groups, i.e. a group-scheme over $V$ -- namely the group-scheme $J$ of regular centralizers. It's defined as follows: for any $x\in \mathfrak g$ take the centralizer of a regular element with the same characteristic polynomial as $x$ (here regular means the dimension of this centralizer is the smallest possible, ie the rank of $\mathfrak g$, or that this centralizer is commutative, or for matrices that minimal polynomial=char.polynomial). Turns out this is a (well-defined) family of commutative groups over $\mathfrak g$.
For $GL_n$ this is a simple classical construction -- $J_x$ is just invertible functions on the spectrum of $x$.
Ngô showed that $J$ acts inertially on $\mathfrak g / G$ (and that the action deloops to an action of the group-stack $BJ$ on $\mathfrak g/G$). More concretely, he showed the tautological isomorphism between $J_x$ and the inertia of the $G$-action (the centralizer $I_x\subset G$ of $x$) for $x$ regular extends uniquely (by Hartogs' theorem) to a map from $J$ to the inertia group-scheme $I$ of $G$ acting on $\mathfrak g$. Moreover this action deloops to an action of $BJ$ on the stack $\mathfrak g/G$.
For $GL_n$ again this is something fairly classical. The inertial action is the statement that invertible functions on the spectrum of a matrix give (by "functional calculus") operators commuting with that matrix. Its delooped version is the statement that you can tensor vector spaces with endomorphisms by invertible modules for functions on the spectrum.
(What was important for Ngô is that the fibers of $J$ are in general not connected, as can be seen already for $x=0$ in $SL_2$, and those component groups are crucial for the geometric interpretation of endoscopy.)
Anyway this action gives an interesting quotient 2-stack --- eg on the regular locus the quotient $(\mathfrak g/G)/BJ$ just looks like the affine variety  $\mathfrak g//G\simeq \mathfrak h//W$ of invariant polynomials (equivalently, generically $\mathfrak g/G$ is isomorphic to $BJ$), but eg for $x=0$ this action is trivial, so we find $J_0$ (the centralizer of a regular nilpotent element in $\mathfrak g$) acting trivially as automorphisms of the stabilizer of $0$ (i.e. $G$ itself), so producing genuine 2-stackiness.
A: Much before Pardon's beautiful result https://arxiv.org/abs/1906.05816 that any compact CW-orbispace (=topological stack) with finite isotropy groups is of the form $[M/G]$ for $G$ a compact Lie group,
I had shown in joint work with David Metzler https://arxiv.org/abs/math/0302182 (Thm 5.5 and 5.7) that it is of the form $[[M/G]/H]$ for $G$ and $H$ compact Lie groups (at least in the case of orbifolds).
