Low-degree mod 2 cohomology of classifying spaces of $G_2$-gauge groups

Let $\mathcal G_k$ be the gauge group of the principal $G_2$-bundle over $S^4$ classified by $k\inπ_4(BG_2)\cong\mathbb Z$. In this paper, we determine the first positive fibre row in the mod $2$ Serre spectral sequence of $Ω^3_0G_2\to B\mathcal G_k\xrightarrow{p_k}BG_2$. The fibre cohomology vanish