For a field $K$ and natural numbers $n, m$, define $A(n,m,K) \in \mathbb{N} \cup \{\infty\}$ as follows: For an $n$-dimensional vector space $V$ over $K$, the group of affine automorphisms of $V$ acts on the set of $m$-element subsets of $V$. Let $A(n,m,K)$ be the number of orbits of this action.
For example, $A(n, 1, K) = 1$ and $A(n, 2, K) = 1$, provided $n \geq 1$.
What is known about the numbers $A(n,m,K)$?