The regular-polytope conjecture for the index-two extensions of L_2(q)

Let $q$ be an odd square prime power. Let $L_2(q)$ be the simple projective special linear group, and let $b$ and $c$ be the outer automorphisms described above, with $L_2(q)\langle b\rangle=P\Sigma L_2(q)$. Regular-polytope conjecture. The group $L_2(q)\langle c\rangle$ is not the group of a regular polytope, whereas $L_2(q)\langle b\rangle$ is the group of a regular polytope of rank $4$. The claim concerns which of the two index-two extensions of $L_2(q)$ arising from the Klein four subgroup of outer automorphisms can occur as automorphism groups of regular polytopes; the preceding examples support it for $q=9$ and $q=25$, while the general case remains open in the supplied text.