Low-degree mod 2 cohomology of classifying spaces of G₂-gauge groups

Let $\mathcal{G}_k$ denote the gauge group of the principal $G_2$--bundle over $S^4$ classified by $k\in \pi_4(BG_2)\cong \mathbb Z$. 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 \[ \Omega^3_0G_2\longrightarrow B\mathcal{G}_k \xrightarrow{\;\mathrm{ev}\;} BG_2 \] and its Serre spectral sequence, we analyze \[ H^s(BG_2;H^t(\Omega^3_0G_2;\mathbb F_2)) \Longrightarrow H^{s+t}(B\mathcal{G}_k;\mathbb F_2) \] in total degree at most $10$. We show that \[ H^j(\Omega^3_0G_2;\mathbb F_2)=0\quad(1\le j\le 4), \qquad H^5(\Omega^3_0G_2;\mathbb F_2)\cong\mathbb F_2, \] so the first positive-degree fibre class is a generator $u_5\in H^5(\Omega^3_0G_2;\mathbb F_2)$. In this range, the only possible Serre differential with source $u_5$ is \[ d_6(u_5)=\varepsilon(k)x_6, \] where $x_6\in H^6(BG_2;\mathbb F_2)$ and $\varepsilon(k)\in\mathbb F_2$. We also prove that, $2$--locally, $\varepsilon(k)$ depends only on $k\bmod 8$, and that $\varepsilon(k)=0$ whenever $8\mid k$.