External wreath product with diagonal action

Definition
This definition uses the left action convention.

Let $$G$$ be any abstract group and $$H$$ be a group along with a homomorphism $$\rho:H \to \operatorname{Sym}(S)$$ for some set $$S$$ (in other words, we are given a permutation representation, or group action, of $$H$$). Let $$K$$ be a group along with a homomorphism $$\alpha:K \to \operatorname{Aut}(G)$$.

Then, the external wreath product with diagonal action of $$G$$ with $$H$$ as the wreathing group (relative to $$\rho$$) and $$K$$ as the diagonally acting group is defined as the external semidirect product of the group $$G^S$$ by $$H \times K$$, where a pair $$(h,k) \in H \times K$$ sends $$f:S \to G$$ to the function $$\alpha(k) \circ f \circ \rho(h^{-1})$$.

The group $$G$$ is termed the base of the wreath product with diagonal action.

The wreath product with diagonal action can be viewed as follows: consider the group $$G^S$$ obtained as a direct product of $$|S|$$ copies of $$G$$ labeled by the elements of $$S$$. The wreath product with diagonal action is the semidirect product of $$G^S$$ by the action of $$H \times K$$, where $$K$$ acts by automorphisms on each coordinate via $$\alpha$$, and $$H$$ permutes the coordinates via the action $$\rho$$ on $$S$$.

Note that in the case that $$K$$ is the trivial group, we simply get the external wreath product of $$G$$ by $$H$$.

Related notions

 * Restricted external wreath product with diagonal action
 * External wreath product
 * Restricted external wreath product