Let $(L,\langle -,-\rangle)$ be an even integral lattice, and let $(A,q)$ be the associated discriminant form: $$ A=L^*/L, \quad q(a)=e^{\pi i \langle a,a\rangle}. $$ We let $\hat L$ to be the extension of $L$ by $\{\pm1\}$ such that $\hat a\hat b=(-1)^{\langle{a,b}\rangle} \hat b\hat a$.

Then we have three natural "isometry groups", $O(L)$, $O(A)$ and $O(\hat L)$; the first two are subgroups of automorphism groups preserving $\langle,\rangle$ or $q$; the last is a natural extension of $O(L)$ by $\mathrm{Hom}(L,\{\pm1\})$.

Let us now consider the modular tensor category $\mathcal{C}(A,q)$ associated to $(A,q)$. This is also the category of modules of the lattice VOA $V_L$ constructed from $L$.

Let $\mathrm{Aut}(\mathcal{C})$ be the group of braided auto-equivalences of a modular category $\mathcal{C}$ up to natural transformations. $\mathrm{Aut}(\mathcal{C}(A,q))$ is known to be equal to $O(A)$.

So we have a natural sequence of group homomorphisms $$ O(\hat L)\to O(L)\to O(A)=\mathrm{Aut}(\mathcal{C}(A,q)) . $$

In [ENO, arXiv:0909.3140], extendsion of a modular tensor category $\mathcal{C}$ by any group $G$ with a homomorphism to $\mathrm{Aut}(\mathcal{C})$ was studied, and two obstructions are identified. The first obstruction takes values in $H^3(G,\mathrm{Inv}(\mathcal{C}))$ where $\mathrm{Inv}(\mathcal{C})$ is the group of invertible objects of $\mathcal{C}$.

So it seems natural to study the obstructions to extending $\mathcal{C}(A,q)$ by any subgroup of $O(A)$, $O(L)$ or $O(\hat L)$. The first obstruction takes values in $H^3(G,A)$. (I believe the first obstruction vanishes for (subgroups of) $O(\hat L)$.)

Does anybody know of any references concerning this point? I would be happy with any partial results, or any starting point for me to explore the literature. (Also, does the extension theory of ENO appear somewhere in the theory of lattice VOA?)