The non-existence of some Galois representations of moderate dimension in small characteristic

Refining arguments of Hyunsuk Moon, under the assumption of the Generalized Riemann Hypothesis, we prove the non-existence of irreducible mod 2 Galois representations unramified outside 2 of dimensions $\le 4$, and of totally real such representations of dimensions $\le 8$. We also prove the non-existence of irreducible totally real mod 3 representations unramified outside 3 of dimensions $\le 4$. We show unconditionally that the image of an irreducible mod 2 symplectic 4-dimensional Galois representation that is unramified outside 2 must be large. Under GRH, we then deduce the non-existence of such representations.