gr.group theory - Automorphism group objects

Consider a monoidal category C with operation $otimes$, unit object $1$, and diagonal map $delta:A to A otimes A$ for all $A in C$ (with naturality conditions on the diagonal map).

We can define a notion of "group object" in the category, where a group object is an object G of C along with a map from $1$ to G (playing the role of the identity element), a map from G to G (playing the role of the inverse map) and a map $G otimes G to G$ (playing the role of the multiplication). Then, we put compatibility conditions on these operations that are commutative diagrams corresponding to associativity, identity elements and inverses. Note that we need to use the diagonal morphism to formulate the condition on inverses.

Wikipedia defines group objects only in the case where the monoidal operation is the categorical product, so if necessary, we can restrict attention to just those cases.

We can define group object morphisms, etc. Some examples of group objects are: topological groups (category of topological spaces with continuous maps and Cartesian product), Lie groups (category of differential manifolds with smooth maps and manifold product), abelian groups (category of groups with group homomorphisms and direct product, by the Eckmann-Hilton principle), and groups (category of sets).

We can then proceed to define a "group object action" of a group object G on an object A by the analogues of the two conditions for a group action. My questions:

1. Under what situations does there exist a group object that plays the role played by the symmetric group on a set? i.e., A group object $operatorname{Sym}(X)$ for each X in C with an action on X such that any group object action of G on X corresponds to a group object homomorphism $G to operatorname{Sym}(X)$ with the desired compatibility conditions?


2. How does this group object $operatorname{Sym}(X)$, if it exists, relate to the automorphism group $operatorname{Aut}_C(X)$ (which is an actual group, not a group object)?


3. We can also define a "group object action by automorphisms" as the action of one group object on another satisfying the additional condition of being group automorphisms. (We need to use the diagonal morphism to formulate this compatibility condition). Is there some object called the "automorphism group object of a group object" such that any group object action corresponds to a group object morphism to the automorphism group object?


4. How does this automorphism group object relate to the automorphism group in the group object category?