I know how to prove the following result. However, my proof is a little bit long and complicated and only uses fairly low tech results in group cohomology. It would be nice if I could find a citation in the literature that directly implies this claim but I am not experienced in group theory or homological algebra and to be honest I'm not sure where to start looking for such a result.



> 
> **Lemma**: Suppose $\mathbb{T}^n = \mathbb{R}^n/ \mathbb{Z}^n $ is the standard $n$-dimensional torus and $\Gamma$ is a finite group of order $k$. Consider a short exact sequence:
> $$ 0 \to \mathbb{T}^n \to E \to \Gamma \to 1. $$
> Suppose $ R := \{ t \in \mathbb{T}^n \mathrel| kt = 0 \} $. Then the following short exact sequence splits:
> $$ 0 \to \mathbb{T}^n/R \to E/R \to \Gamma \to 1. $$
> 
> 
> 


The proof basically boils down to the fact that $ H^2( \Gamma, \mathbb{T}^n) $ is annihilated by $k$ and using that to show that the induced map $ H^2(\Gamma, \mathbb{T}^n) \to H^2(\Gamma, \mathbb{T}^n/R) $ is the zero map and therefore the curvature class of the top short exact sequence projects to zero.


I guess the things I would like to know are:


* Is this 'obvious' to those experienced with finite group extensions or Lie theory?
* Is there a textbook/paper where I can find a result or exercise that directly implies this lemma?



Regarding extensions of finite groups by Tori
(I write an answer rather than a comment in order to accommodate exact sequences.)


Let
$$0\to T\to E\to\Gamma\to 1\tag{$E\_1$}$$
be your first group extension, where $T$ is the torus ${\Bbb R}^n/{\Bbb Z}^n$.
Write $R\_k\subset T$ for the kernel of multiplication by $k$ in $T$
and consider your second exact sequence
$$0\to T/R\_k\to E/R\_k\to\Gamma\to 1.\tag{$E\_2$}$$
To the extension $(E\_1)$ we associate its cohomology class $\eta\_1\in H^2(\Gamma,T)$, and to extension $(E\_2)$ we associate its class $\eta\_2\in H^2(\Gamma,T/R\_k)$.
Then it follows from the constructions of $\eta\_1$ and $\eta\_2$ that $\eta\_2$ is the image of $\eta\_1$ under the homomorphism
$$\phi\_\*\colon H^2(\Gamma,T) \to H^2(\Gamma,T/R\_k)$$
induced by the canonical homomorphism
$$\phi\colon T\to T/R\_k.$$
We have not yet used the assumption that $T$ is a torus and that $\#\Gamma=k$.


Now consider the *surjective* homomorphism
$$\alpha\colon T\to T,\quad x\mapsto kx.$$
Its kernel is $R\_k$, and so it induces an isomorphism
$$\alpha\_\*\colon T/R\_k\to T.$$
Identifying $T/R\_k$ with $T$ using $\alpha\_\*$, we obtain that our
$$\phi\colon T\to T$$
is multiplication by $k$.
It follows that
$$\phi\_\*\colon H^2(\Gamma,T) \to H^2(\Gamma,T)$$
is multiplication by $k$ as well.
Since $\Gamma$ is a group of order $k$, multiplication by $k$ annihilates $H^2(\Gamma,T)$.
See Corollary 1 of Proposition 8 in Section 6, page 105, of:
Atiyah and Wall, Cohomology of groups, in: Cassels and Fröhlich (eds.), Algebraic Number Theory, Acad. Press 1967, pp. 94-115. It follows that $\eta\_2=0$ and the sequence $(E\_2)$ splits.


