We investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general framework to represent pairs of finite groups (G,H), where H⊴G, as pairs of correlation--automorphism groups of suitable incidence geometries. Specifically, we prove that for every such pair (G,H), there exists a finite incidence geometry Γ
Nearby in the stack
satisfying that the pair
(Aut(Γ),AutI(Γ))
of correlation--automorphism groups of
Γ
is isomorphic to
(G,H)
. Our construction proceeds in two main steps: first, we realize
(G,H)
as the correlation and automorphism groups of an incidence system; then, we refine this system into a genuine incidence geometry preserving the same pair of automorphisms groups. We also provide explicit examples, including a family of geometries realizing