Fix a group $G$. We are asked to find interesting functors between $\mathbf{Set}$ and $[G, \mathbf{Set}]$ (the category of left $G$-sets). Here, instead of considering the objects of $[G, \mathbf{Set}]$ to be functors $G \rightarrow \mathbf{Set}$, simply consider them to be pairs $(S, \cdot)$ where $(\cdot) : G \times S \rightarrow S$ satisfies $g_1 \cdot (g_2 \cdot s) = g_1 g_2 \cdot s$.

An obvious start would be the forgetful functor $U : [G, \mathbf{Set}] \rightarrow \mathbf{Set}$ that takes a $G$-set to its underlying set. We ask if this functor has an adjoint. Indeed, let $F : \mathbf{Set} \rightarrow [G, \mathbf{Set}]$ be defined as follows: for a set $S$, $F(S) = (G \times S, \cdot)$ where $g' \cdot (g, s) = (g'g, s)$. For a map of sets $f : S \rightarrow T$, $F(f) : (G \times S) \rightarrow (G \times T)$ is the map $(g, s) \mapsto (g, f(s))$, which is easily verified to be $G$-equivariant. Notice this is essentially a 'free' functor, taking any set $S$ to a $G$-set in the most general arbitrary way.

Let's make this functor $F$ more precise for the sake of pedantry. Let $A$ be a $G$-set and $B$ a set. If $t : U(A) \rightarrow B$ is a map of sets, we define $\overline{t} : A \rightarrow F(B)$ as $\overline{t}(a) = (1, t(a))$. In the other direction, if $r : A \rightarrow F(B)$ is a $G$-equivariant map, define $\overline{r} : U(A) \rightarrow B$ where $\overline{r}(a) = b$ iff $r(a) = (\dash, b)$.

This idea also works for the categories $\mathbf{Vect}_k$ of vector fields over $k$ and $[G, \mathbf{Vect}_k]$ of $k$-linear representations of $G$. Again let $U : [G, \mathbf{Vect}_k] \rightarrow \mathbf{Vect}_k$ be the forgetful functor. This has a left adjoint $F$ where, for a $k$-vector space $V$ with basis $\set{v_1, \dots, v_r}$ and assuming $G = \set{g_1, \dots, g_s}$, we have $F(V) = kG \otimes V$ which has basis $\setbuilder{g_i \otimes v_j}{1 \leq i \leq s, 1 \leq j \leq r}$, equipped with the representation $g \cdot \sum_{i,j} \lambda (g_i \otimes v_j) = \sum_{i,j} \lambda (g g_i \otimes v_j)$. This is the so-called 'induced representation' from representation theory in the special case that the sub-representation is the trivial representation of the trivial subgroup. Again, the 'induced' representation captures the idea of the 'most general' representation that extends another.