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:
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.
What is the quotient of a stack by an algebraic group?
Is there a canonical morphism like the above in the stack world and is it an isomorphism?
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?