On regular polytopes of rank $3$

It is proved that if a finite group $G$ is generated by three involutions $\alpha,\beta$ and $\gamma$, such that $\alpha$ and $\gamma$ commute, and the orders of the products $\alpha\beta$ and $\beta\gamma$ are greater than $2$, then the generating set $\{\alpha,\beta,\gamma\}$ makes $G$ the automorphism group of a regular $3$-polytope if and only if the intersection $\langle\alpha\beta\rangle\cap\langle\beta\gamma\rangle$ contains no non-trivial normal subgroup of $G$, and the intersection $\langle\alpha,\beta\rangle\cap\langle\beta,\gamma\rangle$ is not an elementary abelian subgroup of order $4$. This criterion complements a theorem by M. Conder and D. Oliveros (J. Combin. Theory Ser. A, 2013, v. 120, no. 6, pp. 1291–1304).