It is a well known fact that the irreducible representations of $SL_2(\mathbb{F}_p)$ over $\overline{\mathbb{F_p}}$ are given by the symmetric powers $Symm^k(V)$, where $V = \overline{\mathbb{F}_p}^2$, for $k$ ranging in $0,\ldots,p-1$. In the case of $GL_2(\mathbb{F}_p)$ irreducible representations are exhausted by:

$$\sigma_{k,n} = Symm^k(V) \otimes det^n$$

My question is then if there is a systematic way to know which of these irreducible representations appear in higher symmetric powers ($k>p-1$), either for $SL_2$ or $GL_2$. For example, $Symm^p(V)$ contains a copy of $V$ generated by $x^{\otimes p}$ and $y^{\otimes p}$ with quotient isomorphic to $Symm^{p-2}(V)$.

And what about tensor products $Symm^k(V) \otimes Symm^l(V)$?