Edit: In case that there is no solution for the original question, I modify to enrich the question.


We like to ask a possible specific inflation a $H^3(Q, \mathbb{R} /\mathbb{Z})$ cocycle with a finite group $Q$ into a coboundary in the following two cases in quaternion group or dihedral group:


1. Inflate the 3-cocycle $\alpha\_{1}$ in $Q=Z\_2$ via a dihedral group $G=D\_8$ of order 8.
2. Inflate the 3-cocycle $\alpha\_{2}$ in $Q=Z\_2 \times Z\_2$ via a quaternion group $G=H\_8$ of order 8.


Consider the cocycle $\alpha\_1(g\_a,g\_b, g\_c) \in H^3(Z\_2, \mathbb{R} /\mathbb{Z})$ and $\alpha\_2((g\_{a1},g\_{a2}),(g\_{b1},g\_{b2}),(g\_{c1},g\_{c2})) \in H^3(Z\_2 \times Z\_2, \mathbb{R} /\mathbb{Z})$ in the 3rd cohomology group of $Z\_2$ and $Z\_2 \times Z\_2$ respectively, where $g\_a,g\_b,g\_c \in Z\_2$ respectively, and $(g\_{a1},g\_{a2}),(g\_{b1},g\_{b2}),(g\_{c1},g\_{c2})\in Z\_2 \times Z\_2$ respectively.


Let both $\alpha\_1$ and $\alpha\_2$ to be a cup product form:
$$\alpha\_1(g\_a,g\_b, g\_c)=(-1)^{g\_{a}g\_{b}g\_{c}}.
$$
$$\alpha\_2((g\_{a1},g\_{a2}),(g\_{b1},g\_{b2}),(g\_{c1},g\_{c2}))=(-1)^{g\_{a1}g\_{b2}g\_{c2}}.
$$



> 
> **question: How can we trivialize a $H^3(Z\_2, \mathbb{R} /\mathbb{Z})$'s cocycle $\alpha\_1$ into a coboundary and trivialize a $H^3(Z\_2 \times Z\_2, \mathbb{R} /\mathbb{Z})$'s cocycle $\alpha\_2$ into a coboundary 
>  $$\alpha\_1= \delta \beta\_1$$ 
>  $$\alpha\_2= \delta \beta\_2$$ 
>  in a large group dihedral $D\_8$ and quaternion $H\_8$, respectively, by finding the explicit 2-cochain $\beta\_1$ and $\beta\_2$?** 
> 
> 
> 


Where we can regard the group homomorphism $$D\_8 \to Z\_2 \text{ and } H\_8 \to Z\_2 \times Z\_2. $$ 


1. We can choose either the fact that $D\_8/Z\_4=Z\_2$ or $D\_8/(Z\_2)^2=Z\_2$. We can call $D\_8=G$, and $Z\_4$ or $(Z\_2)^2=N$ normal subgroup, and $Z\_2=Q$ as the quotient group. We all have $G/N=Q$.
2. And we can use the fact that $H\_8/Z\_2=Z\_2 \times Z\_2$, we can call $H\_8=G$, and $Z\_2=N$ normal subgroup, and $Z\_2 \times Z\_2=Q$ as the quotient group. We all have $G/N=Q$.



> 
> In particular, not only the explicit 2-cochain, but also I am interested in finding the relations to Lyndon Hochschild Serre spectral sequence and the $d\_2$ differential, its homomorphism $d\_2:H^1(Q,H^1(N,\mathbb{R} /\mathbb{Z}))\to H^3(Q,\mathbb{R} /\mathbb{Z})$ in the $E\_2$ pages. Or whether it requires other $d\_n$ differentials to determine the inflation of cocycle.
> 
> 
> 


p.s. The original post in ME received little attention so I decide to try MO.



Inflate a finite-group cocycle into coboundary in non-Abelian groups
You can determine the possible inflation of $d$-cocycle in $G$ by lifting to a larger group $G$, from the Lyndon-Hochschild-Serre spectral sequence.
For $H/K= G$, with $BG$ path connected and $\pi\_1(BG)$
acting trivially on $H^\*(K, U(1))$, there is a spectral sequence $\{E^{p,q}\_n, d\_n\}$
with $E^{p,q}\_2 = H^p(G, H^q(K,U(1) ))$. The differential $d\_n$: $E^{p,q}\_n \to E^{p+n,q-n+1}\_n$. And $E^{p,q}\_{n+1}=\frac{\text{Ker} d\_n}{\text{Im} d\_n}$ at $E^{p,q}\_{n}$.


We shall focus on the $d\_2$ differential of $E\_2$ page in LHS spectral sequence 
$$d\_2: E^{p,q}\_2 \to E^{p+2,q-1}\_2$$
$$\Rightarrow d\_2: H^p(Q, H^q(N,U(1) )) \to H^{p+2}(Q, H^{q-1}(N,U(1) )),
$$
$$
 d\_2: H^{d-2}(Q, H^{1}(N,U(1) )) \to H^{d}(Q, H^{0}(N,U(1) ))=H^{d}(Q, U(1)).
$$
If we want to trivialize the $d$-cocycle $\omega\_d \in H^{d}(Q, U(1))$, we
can look for a larger group $G$, where $G/N=Q$ for your $G$. However, all the examples above fail to demonstrate the successful inflation. 


**So the answer to your questions shall be negative.** There may be loopholes, but I could not find any examples either. So this is the best conclusion so far.


