On the restriction of some irreducible mod $p$ representations

For a prime $p,$ let ${\mathbb{F}}_q$ be a finite extension of ${\mathbb{F}}_p\mathrm{.}$ The restriction of an irreducible mod $p$ representation of ${\text{GL}}_2({\mathbb{F}}_q)$ to its subgroup ${\text{GL}}_2({\mathbb{F}}_p)$ can be seen as a tensor product of irreducible representations of ${\text{GL}}_2({\mathbb{F}}_p)\mathrm{.}$ In this paper, we study the restriction of some of these representations of ${\text{GL}}_2({\mathbb{F}}_q)$ to ${\text{GL}}_2({\mathbb{F}}_p),$ for $q=p^2$ and $p^3$ using elementary tools and give explicit socle filtration when $q=4.$ We prove that when $q=p^2,$ a special class of representations of ${\text{GL}}_2({\mathbb{F}}_q)$ are distinguished by suitable characters of ${\text{GL}}_2({\mathbb{F}}_p)\mathrm{.}$