[1]:http://pages.bangor.ac.uk/~mas010/pdffiles/brown-spenmcer-G-groupoid.pdf

The OP asks: How can groupoids be used to describe symmetries in this category? Here are some suggestions for 
starting. 

A principal $G$-bundle $E \to B$ can also be desribed as a groupoid $P= EE^{-1}$ over $B$, a construction due to C. Ehresmann. Here $P(b,c)$ is the set of $G$-maps $E_b \to E_{c}$. So one may ask: what for groupoids generalises the well known inner automorphism map $G \to Aut (G)$ for a group $G$?  

Now the category $Gpd$ of abstract groupoids is cartesian closed, this is one aspect of the utility  of groupoids. We can write the exponential law as a bijection 
$$Gpd(G \times H,K) \cong Gpd(G, GPD(H,K)).$$
The objects of $GPD(H,K)$ are the morphisms, or functors, $H \to K$ and the arrows of $GPD(H,K)$ are the natural equivalences of functors. In the case of groups, these are just conjugacies of morphisms. 



So for any groupoid $G$ there is an endomorphism object $END(G)$ which is a monoid object in groupoids, and this has a maximal subgroup object $AUT(G)$ which is a group object in the category of groupoids. However as shown in the paper available [here][1], group objects in groupoids are equivalent to crossed modules, and the crossed module one obtains by this process is of the form $d: S(G) \to Aut(G)$ where is $S(G)$ is the group of _admissible sections_ $\sigma$ of $G$ as defined by Ehresmann in his paper on topological and differentiable categories. Such a $\sigma $ is a section of say the source map $s$ such that $t\sigma$ is a bijection on $Ob(G)$. These have a multiplication defined by Ehresmann: $\sigma \tau (x)= \sigma (t\tau x) \tau)x)$ (or analogous, depending on conventions conventions). (Such $\sigma$ are called _bisections_ in Mackenzie, K.C.H., _General theory of Lie groupoids and Lie algebroids_, London Mathematical Society Lecture Note Series, Volume 213. Cambridge University Press, Cambridge (2005). )

Can one use this framework to study the differentiable, or Lie, case? 

I may be able to add more later.