Let $G$ denote the $\operatorname{Spin}(n)$ group with $n>4$ and let $\Gamma$ be a cyclic subgroup $G$ of a prime order $p >2$. When does the projection $G \to G/\Gamma$ induce a surjection
between cohomology groups $H^3$ with integral coefficients?



Cohomology of the quotient of a Lie group by a finite subgroup
Here is a solution for $G$ a compact, simple, connected, simply-connected Lie group and $\Gamma$ a subgroup of the center of $G$.


The group $H^3(G,Z)$ classifies $U(1)$-gerbes over $G$. A gerbe $\mathcal{G}$ is in the image of the pullback map $$H^3(G/\Gamma,Z) \to H^3(G,Z)$$ if and only if it admits a *$\Gamma$-equivariant* structure. Indeed, in this case one can form the quotient gerbe $\mathcal{G}'$ over $G/\Gamma$, and the pullback of $\mathcal{G}'$ is isomorphic to $\mathcal{G}$.


The crucial point is that the basic gerbe $\mathcal{G}^1$, i.e. the one that represents a generator of $H^3(G,Z)=Z$, enjoys an explicit, Lie-theoretical construction in the framework of *bundle gerbes*. The existence of $\Gamma$-equivariant structures can then be checked by inspection of a certain obstruction class.


This obstruction class has been computed explicitly for all possible *central* subgroups. For example, the obstruction for the gerbe $\mathcal{G}^k$ over $SU(n)$ (which represents $k \in H^3(SU(n),Z)$) vanishes if either $k$ is even, or $|\Gamma|$ is odd, or $\frac{n}{|\Gamma|}$ is even. That's how I came up with the comment to the question.


All constructions and calculations are in: K. Gawedzki and N. Reis "Basic gerbe over non simply connected compact groups" J. Geom. Phys., 2003, 50, 28-55. An overview is in Table 5.1 on page 143 of my [phd thesis](http://www.konradwaldorf.de/docs/waldorf_phd.pdf) ([Wayback Machine](https://web.archive.org/web/20220423130515/http://www.konradwaldorf.de/docs/waldorf_phd.pdf)).


EDIT: For non-central groups one can still use the basic gerbe $\mathcal{G^1}$, but I don't know if the obstruction classes are then still accessible for calculations.


