Transitive families of transformations

In the first part of the paper we model an abstract version of Sabinin's theory on transitive families $\mathcal S$ of diffeomorphisms on a differentiable manifold, in particular we define an abstract holonomy group. In the second part we determine the linear connection associated with a smooth family $\mathcal S$ and clarify the relations between it and the properties of $\mathcal S$. Moreover, we prove that all natural holonomy groups are isomorphic, if $\mathcal S$ is a geodesic system. Finally we show that the group $\mathcal A$ of smooth automorphisms of $\mathcal S$ is a Lie subgroup of the group of affine transformation of the underlying manifold of $\mathcal S$; if $\mathcal A$ acts transitively we enlighten how $\mathcal A$ influences the algebraic as well as the differential geometric properties of $\mathcal S$.