Let $A$ be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form $langle,rangle : A times A to mathrm{U}(1)$. Then you can reasonably talk about the “orthogonal group” $O(A,langle,rangle)$, i.e. the automorphisms of $A$ which preserve $langle,rangle$.

Actually, when $A$ has even order, the most standard version of “orthogonal group” is to choose a quadratic refinement $q : A to mathrm{U}(1)$ of $langle,rangle$, i.e. a quadratic function such that $q(a_1 a_2) / q(a_1)q(q_2) = langle a_1,a_2rangle$, and to ask for the group $O(A,q)$ of automorphisms preserving $q$. (When $A$ has odd order, there is a unique quadratic refinement, and so the two notions agree. In general, the quadratic refinements form a torsor for $hom(A, mathbb{Z}/2)$.) I can ask my question for both versions of $O(A)$, but in fact the one I care about is the “nonstandard” $O(A,langle,rangle)$ when $A$ has order a power of $2$.

**I am interested in understanding the low $mathbb{Z}/2$-cohomology of $O(A,langle,rangle)$.**

I specifically want to know about classes which are “stable”, but I am having trouble saying precisely what I want “stable” to mean. Certainly if I have two groups $A, A’$ each equipped with nondegenerate symmetric bilinear forms, then I only care about the image of restriction along $O(A) subset O(A) times O(A’) subset O(A times A’)$. But I expect that if there is a stronger notion of “stability”, then I only care about those classes.