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Let $G$ be a group, actiong on a set $X$ and $H$ a normal subgroup. Then we have a canonical isomorphism $$(X/H)/(G/H)\rightarrow X/G$$ I would like to have a statement like this for stacks, more precisly I'm interested in the situation where $X$ is a variety and $G$ is a linear algebraic group. So I have the following questions:

  1. To make sense of the above statement, the first thing we need to know is, what it means for an algebraic group ($G/H$ in our case) to act on a stack. So what would be the correct definition for the action of a group on a stack.

  2. What is the quotient of a stack by an algebraic group?

  3. Is there a canonical morphism like the above in the stack world and is it an isomorphism?

  4. To me the above statement looks like a very natural question to me, but I didn't find any discussion in literature, is there a reference?

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There is a paper by Matthieu Romagny, "Group actions on stacks", in the Michigan Journal of Mathematics http://www.math.jussieu.fr/~romagny/articles/group.pdf, where the first two points are discussed in detail. I don't remember if he proves the isomorphism you want, which holds for sure, and is not hard to establish

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