From the question https://www.physicsoverflow.org/32208, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action

$$I(g)=\frac{1}{4\pi}\int_{M}\mathrm{Tr}(\omega\wedge d\omega+\frac{2}{3}\omega\wedge\omega\wedge\omega)$$

i.e. it changes under a twist of framing on $M$ by $I(g)\rightarrow I(g)+2\pi s$ with $s\in\mathbb{Z}$, is related with the group $H_{3}Spin(3)=\mathbb{Z}$.

1. What is this group $H_{3}Spin(3)$?

2. Why is it isomorphic to $\mathbb{Z}$?

3. How exactly is it related with the change of Pontryagin class under a change of framing on $M$?

They also talked about $\Omega_{3}^{fr}=\mathbb{Z}_{24}$.

4. What exactly is this $\Omega_{3}^{fr}$?