Let $G$ be a group, and consider the action of $G$ on itself by conjugation. If we think of $G$ as a one object category, then the conjugation action can be realised as automorphisms of this category, and we may build the associated $2$ category with one object, with additional $2$ morphisms given by elements of $G$, acting as conjugation.


The question is whether one can extend this construction to take into account that the $2$ morphisms also have a notion of "equivalence", whether they are conjugate when viewed as elements of $G$. These relations should be witnessed by elements of $G$, expressing when two $2$ morphisms are conjugate, which then have more relations witnessed by elements of $G$ (by conjugation on these witnesses), and so on.


Ideally this whole package would respect the underlying group structure in some sense, since we are considering automorphisms of all the data in the previous stage to obtain the next stage.


One can give silly ways of describing this heap of data, so as a test/benchmark, one could ask for some general categorical object, which is built only out $G$ as a one object category, which observes "categorically" the following fact about finite groups: For $G$ finite, with $p\in \mathbb{Z}$ coprime to $|G|$, then for any $x,y\in G$, there exists $g,h\in G$ with $(xy)^p=gx^p g^{-1}h y^p h^{-1}$.



Is there a higher categorical structure which models the (higher) conjugation actions of a group acting on itself?
Following up a bit on Fernando's comment, you look to be trying to define things a bit like the tensor square of a group (acting on itself). Look at the work by Ronnie Brown, et al, *Some computations of non-abelian tensor products of groups*, J. Algebra, 111, (1987), 177 – 202. This would give you a crossed square. There are variants using an exterior product, which may also be of interest (see work by Graham Ellis)


As Fernando mentions there are $cat^n$-groups / crossed n-cubes of groups that might encode higher commutator data but I think you would be needing more than just a single group, for instance a group with a family of normal subgroups.


There is a separate response, which says that if G is thought of as a one object groupoid, then there is the functor category $G^G$, which encodes the conjugation. It has an associated crossed module which is exactly the $G\to Aut(G)$ one that was mentioned by Fernando.


