A Chinese remainder theorem for Galois representations

For any list of not necessarily distinct quartic polynomials
$g_1, … , g_n$ with non-zero constant terms $(\mod 2)$ and any list of distinct primes $q_1, … , q_n$ all greater than 53, we construct infinitely many surjective Galois representations $\rho: G_Q \to GL_4(F_2)$ where the characteristic polynomial of $\rho$ at $Frob_{q_i}$ is $g_i$ for all $i$. Using the effective Chebotarev density theorem for function fields and geometric Galois extensions of $Q(t)$ with $GL_d(F_q)$-Galois groups, we reduce this to a problem about geometric algebra and permutation groups; the exceptional isomorphisms Alt(8) = $GL_4(F_2)$ = $P\Omega_6^{+}(F_2)$ play a key role. We also have similar results for other residual representations.