Sharp Fourier restriction theory and finite field extension theory have both been topics of interest in the last decades. Very recently, in GonzalezOliveira, the research into the intersection of these two topics started. There it was established that, for the (3, 1) -cone Γ (₃, ₁) ³: =\η Fq⁴\{0\: η₁²+η₂²+η₃²=η₄²\}, the Fourier extension map from L² L^4 is maximized by constant functions when q=3\, 4. In this manuscript, we advance this line of inquiry by establishing sharp inequalities for the L^2 L^4 extension inequalities applicable for all remaining cones Γ³ Fq⁴. These cones include the (2, 2) -cone Γ (₂, ₂) ³: =\η Fq⁴\{0\: η₁²+η₂²=η₃²+η₄²\} for general q=pⁿ and the (3, 1) -cone when q=1\, 4. Moreover, we classify all the extremizers in each case. We note that the analogous problem for the (2, 2) -cone in the euclidean setting remains open.
González-Riquelme et al. (Sat,) studied this question.