I started reading Non abelian differential gerbes https://arxiv.org/abs/math/0511696v5
It says in abstract :
We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids.
It says in section 2, named Differentiable gerbes as groupoid extensions that,
The purpose of this section is to set up the basic notions of differentiable gerbes in terms of Lie groupoid extensions.
Then they start discussing what is a Lie groupoid extension and all but I was not able to find any definition for what does it mean to say differentiable gerbe or how they are relating differentiable gerbes with Lie groupoid extensions.
There is only one remark in page $8$ saying
There is a 1-1 correspondence between Morita equivalence classes of Lie groupoid extensions and (equivalence classes of) differentiable gerbes over stacks.
But I do not see any reference given for this remark.
Can some one help me to understand how differentiable gerbes(I know them as stacks over a manifold that are locally non empty and locally connected) are related to Lie groupoid extensions.
More over, what does it mean to say differentiable gerbes over stacks? Does it mean something to do with morphism of stacks where domain is a gerbe?
Are there other places where Lie groupoid extensions are discussed in detail?
Question : Can some one provide a reference for seeing Differentiable gerbes on stacks as Lie groupoid extensions.
I remember seeing gerbes on Manifolds are some how related to some crossed modules extensions. Could not recall or find the reference for that as well.