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, 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.