How to reconstruct geometric object that a Frobenius group acts on

Question

A Frobenius group has equivalent definitions:

1. a transitive permutation group on a finite set such that no non-trivial element fixes more than one point and some non-trivial element fixes a point.

2. has nontrivial malnormal subgroup, $H$, which acts (as a semidirect product) on a normal subgroup $N$, with each element fixing only the identity.

Question: Given a group $G=N\rtimes H$ that is Frobenius by definition 2 (i.e. knowing the isomorphism classes of $H$ and $N$ and the $\varphi:H\rightarrow \operatorname{Aut}(N)$ defining the semidirect product), how can I reconstruct the set in definition $1$ that $G$ is supposed to act on?

My question was motivated by reading user35603's answer which illustrates how to view $C_p\rtimes C_2$ (with $p$ odd) as a Frobenius group using the first definition. It is very visual and we can see that it is Frobenius by #1. Can our reconstruction be similarly geometric and intuitive?