On non-admissible irreducible modulo $p$ representations of $\protect \mathrm{GL}_{2}(\protect \mathbb{Q}_{p^{2}})$

We use a Diamond diagram attached to a 2-dimensional reducible split mod $p$ Galois representation of $\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p^{2}})$ to construct a non-admissible smooth irreducible mod $p$ representation of $\mathrm{GL}_{2}(\mathbb{Q}_{p^{2}})$ following the approach of Daniel Le.