  We decompose the tensor product of two irreducible representations of
$\mathrm{GL}_2(\mathbb{F}_q)$ for odd $q$ and classify the pairs such that
their tensor product is multiplicity free. We also classify the pairs such that
their tensor product has unique decomposition property. We additionally
characterize the self-dual irreducible representations of
$\mathrm{GL}_2(\mathbb{F}_q).$
