$\newcommand{\SH}{\mathit{SH}}\newcommand{\Z}{\mathbb Z}$Let $G$ be a compact Lie group and $\lambda\in
H^4(BG;\Z)$. The data $(G, \lambda)$ determine a 3d topological field theory called Chern-Simons theory, except not
quite: there is an obstruction to defining it on general closed, oriented $3$-manifolds called the *anomaly*.
Freed-Teleman characterize anomalies of $n$-dimensional field theories as
$(n+1)$-dimensional invertible field theories, which have been classified. I think in this case the anomaly
field theory should be unitary, so is classified up to isomorphism by a torsion element of
$\mathrm{Hom}(\Omega_4^{\mathrm{SO}}, \mathbb C^\times)$, by a theorem of
Freed-Hopkins. If I choose $G$ and $\lambda$, is the isomorphism type of the
anomaly field theory, as a bordism invariant, known?

I'm actually interested in a slightly more general story, where $\lambda\in\SH^4(BG)$. (Here $\SH$ is a generalized
cohomology theory called *supercohomology*: $\pi_0\SH = \Z$, $\pi_1\SH = \Z/2$, and the $k$-invariant is nonzero.
When $G$ is simple and simply connected, using $\SH$ instead of $H$ amounts to choosing a half-integer rather than
an integer.) Then there is a 3d spin TFT called spin Chern-Simons theory, which is again anomalous. Now the anomaly
is a torsion element of $\mathrm{Hom}(\Omega_4^{\mathrm{Spin}}, \mathbb C^\times)$.

**For spin Chern-Simons theories, is the isomorphism type of the anomaly known? If not, is there an explicit
conjectured description?**

I'm primarily interested in the spin case when $G$ is a torus, but any information (e.g. $G$ simple and simply connected, only for the oriented case, etc.) is helpful.