Let $E/F$ be a quadratic unramified extension of local fields with $\operatorname{char} F = 0$. Let $\Lambda \subseteq E^2$ be an $O_E$-lattice of rank $2$. Let $g \sigma \in \operatorname{GL}_2(E)$ be a conjugate-linear map where $\mathrm{id} \ne \sigma \in \operatorname{Gal}(E/F)$ is the Galois conjugation.
Given integers $m \le n$, I'd like some reference or statement that describes the set $$\{\text{lattices } \Lambda \subseteq E^2 \mid (\Lambda : g \sigma(\Lambda)) = (m,n)\}$$ where $\pi^m \mid \pi^n$ are the elementary divisors of the pair $\Lambda, g \sigma(\Lambda)$. This probably involves the square $(g \sigma)^2 = g \sigma(g)$ so that this set descents to $O_F$-lattices.
This is related to the theory of Bruhat-Tits buildings for $\operatorname{GL}_2$, and I want to apply this to calculating twisted orbital integrals for $\operatorname{GL}_2$.
