Extended Weil representations: the non-dyadic local field cases

Let F be a non-archimedean local field of odd residual characteristic. Let W be a symplectic vector space over F. It is known that there are different Weil representations of a Meteplectic covering group Mp(W). By some twisted actions, we reorganize them as a representation of $PGMp^{\pm}(W)$, which is a covering group related to the projective similitude symplectic group.