On the restriction of some irreducible mod-p representations of GL₂ (Fq) to GL₂ (Fₚ)

For a prime p, let Fq be a finite extension of Fₚ and G=GL₂ (Fq). Then the irreducible representations of G are classified as twists of Sym^r (F₏ℂ). The restriction of irreducibles of G to its subgroup G=GL₂ (Fₚ) is same as investigating the behavior of the tensor product of irreducible representations of G. In this paper, we study the restriction of some of these representations of G to G, for 2 and 3 degree extensions of Fₚ.