Let $\Gamma (G;(G_i)_{i \in I})$ be a coset geometry (in the sense of Buekenhout) firm, residually connected and flag transitive with Borel subgroup $B$. Consider $H$ any subgroup of $B$. I want to find all the elements $\rho_i$ in the minimal parabolics subgroups $G_{I-\{i\}}$ such that $[ \langle H ,\rho_i \rangle : H] = 2$.

I have been given an idea: consider the quotient $G_{I-\{i\}}/B$ together with the canonical homomorphism $\phi_i : G_{I-\{i\}} \to G_{I-\{i\}}/B$. Then keep all the elements $\rho_i \in G_{I-\{i\}}-B$ such that $\phi_i(\rho_i)$ is an involution. In order to find them, take all involutions in $G_{I-\{i\}}/B$, then take their preimage by $\phi_i$ and it then suffices to multiply the preimage by the elements in the kernel of $\phi_i$ and keep those which normalize $H$ to get the elements satisfying the condition given above.

However, I have two problems. First, I don't really see why this works. Second is that I have no guarantee that $B$ is normal in each minimal parabolic subgroup. Worse, most of the time it is not. Hence, the set of right cosets of $B$ in $G_{I-\{i\}}$ does not have a group structure. Therefore, I have some problem with this solution.

Does anyone have a suggestion?

Thanks in advance.