I am interested in finding mathematical examples and criteria of inflating $Q$-cocycle to coboundary in $H$, under the requirement:

(1) Both $H$ and $Q$ are connected topological groups or Lie groups (i.e. continuous groups). It is even better both are compact groups.

(2) In terms of a group extension $1 \to N \to H \overset{\varphi}{\to} Q \to 1$, I also require that $N$ is a finite group (i.e. discrete groups). Again $H$ and $Q$ are connected topological groups or Lie groups.

(3) We consider the group cohomology and topological cohomology together, by the fact that we can view the classifying space $BQ$ of $Q$ as a topological group. For example, the group cohomology of $SO(3)$ is the cohomology of the topological space $BSO(3)$. We ask that whether a $Q$-cocycle $$\omega^Q_d(q) \in \mathcal{H}^d(Q,\mathbb{R}/\mathbb{Z})$$ can be inflated in $H$ to be a coboundary $$\omega^H_d(h)\equiv\omega^Q_d(\varphi(h))=\delta (\beta_{d-1}(h)) =1\in \mathcal{H}^d(H,\mathbb{R}/\mathbb{Z})$$ so that $\beta_{d-1}(h)$ is a split cochain in a lower dimension, here $d-1$. Here $q \in Q$ and $h \in H$ are in multiplet forms. Here $1\in \mathcal{H}^d(H,\mathbb{R}/\mathbb{Z})$ means it is in $\mathbb{R}/\mathbb{Z}$ value, and which is 0 as the identity element of cohomology group.

My questions are basically:

Examples: What are some more valid examples that satisfy (1), (2), (3)? I am interested mainly in the case $d=2,3,4,5$.

Criteria: What are some effective criteria to find new examples? (e.g. Serre spectral sequence, or topological covering map constraint, differential maps in $E_2$ page, etc.)

The example I already know is that when $d=2$, for $Q=SO(3)$, $H=SU(2)$ and $Q=\mathbb{Z}_2$, we can inflate the 2-cocycle $\omega^Q_2(q) \in \mathcal{H}^2(SO(3),\mathbb{R}/\mathbb{Z})=\mathbb{Z}_2$ to a 2-coboundary when we pull back $SO(3)$ to $SU(2)$.

What other examples can we look for?

The answer will be awarded to the guru who can provide nice examples for illustrations.

Thank you for the patience reading my first post. Hope to get some nice replies! ; )