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.

1. What are examples of $H$ acting on $V/G$ (1a. acting weakly but not strongly) (1b. acting strongly)?
2. What is $(V/G)/H$ in these examples?
3. In general, when is this double quotient a quotient stack, i.e. of the form $V/G'$?