Modular representations arising from self-dual $\mathrm{\ell }$-adic representations of finite groups

Suppose $\mathrm{\ell }$ is a prime number, ${\mathbf{Q}}_{\mathrm{\ell }}$ is the field of $\mathrm{\ell }$-adic numbers, ${\mathbf{F}}_{\mathrm{\ell }}$ is the finite field of $\mathrm{\ell }$ elements, and $d$ is a positive integer. Suppose $G$ is a finite subgroup of a symplectic group $S{p}_{2d}({\mathbf{Q}}_{\mathrm{\ell }})$. We prove that $G$ can be embedded in $S{p}_{2d}({\mathbf{F}}_{\mathrm{\ell }})$ in such a way that the characteristic polynomials are preserved (mod $\mathrm{\ell }$), as long as $\mathrm{\ell }>3$.