The situation in odd characteristic is complicated by the quadratic character χ2: the ! hypergeometric sheaf of type (7, 1)
F(χ2,k) := (A–7)deg ⊗H(, , ,χ2,χ2,χ2,χ2; χ2),
though lisse on Gm, is not pure of weight zero, nor is its Ggeom the group G2. One knows [Ka-ESDE, 8.4.7] that this sheaf sits in a short exact sequence
0 → V ⊗Lχ2 → F(χ2,k) → (A–7)deg ⊗ Kl(, , ,χ2,χ2,χ2)(–1) → 0,
V := ...