Consider the direct product $G \times G$, where $G$ is a finite group. There is a map $s:G\times G\to G\times G$ which switches the components: $(g,h)\mapsto (h,g)$, which is clearly an isomorphism of groups, or even an action of $S_2$ on $G\times G$.
The generalization to many factors is obvious.
Has this been studied? Does this have a name? Is there a reference?
As discussed in comments, what you're looking for is in fact the wreath product. Suppose we are given a finite set $A$, a group $H$ acting on $A$ and any group $G$. Let $G^A$ be the direct product of $|A|$ copies of $G$, which we consider as the group of functions $f:A\to G$ with pointwise multiplication.
The wreath product $G\ \mathrm{wr}_A H$ is the semidirect product $G^A\rtimes H$ where for $h\in H$ and $f\in G^A$ we have $$hf(a)h^{-1}=f(h^{-1}a)$$ This allows us to study the interaction of the group $H$ acting on $A$ with the direct product of $G$ indexed by $A$, permuting the factors as in your question. See the article for examples and literature.