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.

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.