Sharp homogeneity in affine planes, and in some affine generalized polygons

Let G be a collineation group of a generalized (2n + 1 )-gon Γ and let L be a line such that every symmetry σ of any ordinary (2n + 1 )-gon in Γ containing L with σ(L) = L extends uniquely to a collineation in G. We show that Γ is then a Desarguesian projective plane. We also describe the groups G that arise. As a corollary, we treat the analogous problem without the restriction σ(L) = L.