If $X$ and $Y$ are finite transitive $G$-sets, and $G \circlearrowright X \cong G \circlearrowright [G:H]$ and $G \circlearrowright Y \cong G \circlearrowright [G:K]$, then I believe that the $G$-maps $X \to Y$ are in bijection with
$$\text{Cosets }gK \in [G:K] \text{ such that } gKg^{-1} > H$$
Is there a name for this thing? Is it something else in disguise?
Edit: I guess I could say these are cosets of $K$ that comprise full equivalence classes in in $[G:N_GK] \cong \{\text{conjugate subgroups of }K\}$. Namely, the ones where the conjugate subgroup contains $H$. So in particular, the number of such maps is a multiple of $\#[N_GK : K]$.
If $H$ is a subgroup of $G$, then the $G$-set $G/H$ represents the functor of $H$-fixed points: that is, $G/H$ is the free $G$-set on an $H$-fixed point. Hence $\text{Hom}(G/H, G/K)$ is the set of fixed points of the action of $H$ on $G/K$. I think this is the cleanest way to say it.