I'm taking a first course in group representation theory, and one of my earliest problems was to show that if $(\rho, V)$, $(\sigma, W)$ are representations of a group $G$, then $(\tau, \mathrm{Hom}_k(V, W))$ is also a representation where
$$\tau(g)(\varphi) = \sigma(g) \circ \varphi \circ \rho(g^{-1}) \quad \forall g \in G, \varphi \in \mathrm{Hom}_k(V, W).$$
This seems like a natural representation to consider to me, although I'm struggling to find a name for it in my lecture notes or via web searches. Is there a name for this type of representation?
I have checked a few books on representation theory, and no name is mentioned there. But maybe this is no surprise: When $G=1$, so we are just talking about vector spaces, there is also no well-known name for the vector space $\mathrm{Hom}_k(V,W)$. It is just a very basic construction. You could call it the "hom vector space", but it's more common to just describe it as the "vector space of homomorphisms" or linear maps (from $V$ to $W$). Accordingly, for representations, this representation can be called the "hom representation" or the "representation of linear maps" (from $V$ to $W$). As explained in SE/4778476, both constructions are instances of an internal hom, but I doubt that people outside of category theory like to call this representation the "internal hom of representations" (even though it is).