On three families of extended tilde geometries

Let ΓΓ be an extended tilde geometry of rank n>2n>2 such that there exists a 1-covering γ:Γ→Φγ:Γ→Φ where ΦΦ is a c.Cn−1c.Cn−1-geometry with orders 1,2,…,21,2,…,2. Suppose that the normalizer in Aut(Γ) of the deck group of γγ acts flag-transitively on ΓΓ. We prove that, under these hypotheses, only three possibilities exist for the universal cover of ΓΓ.