I'm learning some representation theory in the context of a mathematical physics class. It seems that the main motivation for studying representation theory in physics is the following result: if $G$ is a group which acts on a set $X$, then this extends uniquely to a representation $\pi$ on the space of functions $\{f : X \rightarrow Y\}$ via:
$$[\pi(g)f](x) = f(g^{-1}x).$$
The simplest example of this in physics being translation operators $[\mathcal{T}_{\vec{r}} f](\vec{x}) = f(\vec{x} - \vec{r}),$ where $\vec{x} \mapsto \vec{x} + \vec{r}$ is a group action on $\mathbb{R}^3$.
It seems that in physics it's standard to regard the group action on $X$ and the representation $\pi$ as being identical mathematical objects as an abuse of terminology. My question is: is there a name for the representation (or a theorem stating the existence of) $\pi$ induced on a function space from the group action on the domain?