Suppose $\mathcal{G},\mathcal{H}$ are Lie groupoids and $F:B\mathcal{G}\rightarrow B\mathcal{H}$ be a morphism of stacks.

We can talk about the fibered product $B\mathcal{G}\times_{B\mathcal{H}}B\mathcal{G}$.

Is it true that this stack is also of the form $B\mathcal{K}$ for some Lie groupoid $\mathcal{K}$?

We can ask the same question in the set up of Lie groups. I guess answer would be same in both cases.

Suppose $G,H$ be Lie groups and $F:BG\rightarrow BH$ be a map.

We can talk about the fibered product (pullback) $BG\times_{BH}BG$.

Is it true that this space is also of the form $BK$ for some Lie group $K$?

Suppose we have a map of Lie groups $G\rightarrow H$ inducing the map $F:BG\rightarrow BH$ then, I am guessing that $G\times_HG$ (if it is a Lie group) is the group $K$ I am asking above.

Any comments are welcome.