Weil representations over finite fields and Shintani lift

Let Sp_V(F) be the group of isometries of a symplectic vector space V over a finite field F of odd cardinality. The group Sp_V(F) possesses distinguished representations--- the Weil representations. We know that they are compatible with base change in the sense of Shintani for a finite extension F'/F. The result is also true for the group of similitudes of V.