Let $F$ be an imaginary quadratic field and $𝒪$ its ring of integers. Let $𝔫\subset 𝒪$ be a non-zero ideal and let $p\>5$ be a rational inert prime in $F$ and coprime with $𝔫$. Let $V$ be an irreducible finite dimensional representation of ${\overline{𝔽}}_p\left[{\mathrm{GL}}_2\left({𝔽}_{p^2}\right)\right]$. We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $V$ already lives in the cohomology with coefficients in ${\overline{𝔽}}_p\otimes de{t}^e$ for some $e\ge 0$; except possibly in some few cases.