Tensor power of a group

Definition
Suppose $$G$$ is a group and $$n$$ is a positive integer. The $$n^{th}$$ tensor power of $$G$$, which we may denote as $$\bigotimes^n G$$, is defined inductively as follows. Note that at each stage, the $$n^{th}$$ tensor power has the structure of a crossed module over $$G$$, and it is important to store this structure in order to induct rather than simply storing the group as an abstract group.

Base case
The first tensor power of $$G$$ is $$G$$ itself. The crossed module structure is as follows: the action by automorphisms is the action by conjugation, and the homomorphism is the identity map.

Inductive step
Suppose we have defined $$\bigotimes^n G$$ as a crossed module over $$G$$. Then, define $$\bigotimes^{n+1} G$$ as follows:


 * First, note that since crossed module defines a compatible pair of actions, we have a natural choice of a compatible pair of actions between $$\bigotimes^n G$$ and $$G$$.
 * As an abstract group, define $$\bigotimes^{n+1} G$$ as the tensor product $$\bigotimes^n G \otimes G$$ for this compatible pair of actions.
 * Now use the fact that tensor product of groups has a crossed module structure with respect to each group to give $$\bigotimes^{n+1} G$$ the structure of a crossed module over $$G$$.