The Mathieu group $M_{12}$

Let $G$ be a finite simple non-Abelian group. $t$ is an involution of $G$, and $L=O^2(C_G(t)/O(C_G(t)))$. If the center $Z(L)$ is cyclic and $L/Z(L)\simeq PGL(2,q)$, $q$ odd, then either a Sylow 2-subgroup of $G$ is semidihedral or $C_G(t)\simeq Z_2\times PGL(2,5)$ and $G$ is isomorphic to the Mathieu group $M_{12}$ of degree 12.