Rings of cooperations for hermitian K-theory over finite fields

We compute the ring of cooperations $\pi_{*,*}^{\mathbb{F}_q}(\text{kq} \otimes \text{kq})$ for the very effective hermitian K-theory over all finite fields $\mathbb{F}_q$ where $\text{char}(\mathbb{F}_q) \neq 2.$ To do this, we use the motivic Adams spectral sequence and show that all differentials are determined by the integral motivic cohomology of $\mathbb{F}_q$.

As an application, we compute the $\mathrm{E}_1$-page of the kq-resolution.