# Representations and Group Actions

I have some finite set $$S$$ and a collection of complex vector spaces $$\{W^{s}\}_{s\in S}$$ which are distinguished from one another. I also have a finitely generated group $$G$$ and some left action of $$G$$ on $$S$$ (given some $$g\in G$$ and $$s\in S$$, I'll denote said action by $$g(s)$$ for shorthand). I then have $$|S|$$ unitary representations of $$G$$ that behave as follows:

$$\rho_{s}:G\to{\rm Hom}\big(W^{s},\bigoplus_{g\in G} W^{g(s)}\big)$$ for each $$s\in S$$, where

$$\rho_{s}(g): W^{s}\to W^{g(s)}$$

Of course, $$\dim(W^{g(s)})=\dim(W^{s})$$ for all $$g\in G, s\in S$$ (hence admitting such unitary representations). For instance, given $$g,h\in G$$, then e.g.

$$\rho_{s}(gh)=\rho_{h(s)}(g)\circ\rho_{s}(h)$$

Question:

Has the above setup been studied? What is the appropriate nomenclature?

Edit:

Upon reflection, this seems to be some deconstruction of the representation of a groupoid.