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

Let ${\mathcal{G}}_k$ denote the gauge group of the principal $G_2$–bundle over $S^4$ classified by $k\in {\pi }_4(B{G}_2)\cong \mathbb{Z}$. Motivated by the $p$–local homotopy classification of these gauge groups, due to Kishimoto–Theriault–Tsutaya and Kameko, we study the low-degree mod~$2$ cohomology of the classifying spaces $B{\mathcal{G}}_k$ as unstable modules over the Steenrod algebra. Using the evaluation fibration $${\mathrm{\Omega }}_0^3{G}_2⟶B{\mathcal{G}}_k\stackrel{\mathrm{e}\mathrm{v}}{\to }B{G}_2$$ and its Serre spectral sequence, we analyze $$H^s(B{G}_2;H^t({\mathrm{\Omega }}_0^3{G}_2;{\mathbb{F}}_2))⟹H^{s+t}(B{\mathcal{G}}_k;{\mathbb{F}}_2)$$ in total degree at most $10$. We show that $$H^j({\mathrm{\Omega }}_0^3{G}_2;{\mathbb{F}}_2)=0(1\le j\le 4),H^5({\mathrm{\Omega }}_0^3{G}_2;{\mathbb{F}}_2)\cong {\mathbb{F}}_2,$$ so the first positive-degree fibre class is a generator $u_5\in H^5({\mathrm{\Omega }}_0^3{G}_2;{\mathbb{F}}_2)$. In this range, the only possible Serre differential with source $u_5$ is $$d_6(u_5)=\epsilon (k)x_6,$$ where $x_6\in H^6(B{G}_2;{\mathbb{F}}_2)$ and $\epsilon (k)\in {\mathbb{F}}_2$. We also prove that, $2$–locally, $\epsilon (k)$ depends only on $k8$, and that $\epsilon (k)=0$ whenever $8\mid k$.