Rings of cooperations for hermitian K-theory over finite fields

We compute the ring of cooperations ${\pi }_{\ast ,\ast }^{{\mathbb{F}}_q}(\text{kq}\otimes \text{kq})$ for the very effective hermitian K-theory over all finite fields ${\mathbb{F}}_q$ where $\text{char}({\mathbb{F}}_q)\mathrm{\ne }2.$ 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.