I believe there is a reasonable notion of $\text{Ext}^1(G,H)$ in the category of groups (where $G$ and $H$ are groups). Is there a decent reference describing this?


My particular situation involves a nilpotent Lie group $G$ with finitely many components. One may form the short-exact sequence $$1\rightarrow G\_0\rightarrow G\rightarrow\pi\_0(G)\rightarrow 1,$$ an element of $\text{Ext}^1(\pi\_0(G),G\_0)$. I am curious about conditions under which this sequence splits. I would appreciate any and all references and suggestions.


Thanks! 



Extensions of Groups
Since $G$ is a Lie group with finitely many components, it has a maximal compact subgroup $K$, unique up to conjugation and $G=KG\_0$ (Mostow). Since $G$ is actually nilpotent, $K$ is unique and actually consists of the elements in $G$ contained in a compact subgroup. In particular $K$ is normal in $G$. 


From $G$ we can define $K$ as above; conversely if we know $K$ and $G\_0$, we know $G$, which is the quotient of $K\times G\_0$ by the central subgroup $\{(g,g^{-1}),g\in K\_0\}$.


This shows that if $F$ is a finite nilpotent group, the set classifying extensions of $G\_0$ by $F$ that are nilpotent is in canonical bijection with the set classifying central extensions of $K\_0$ by $F$. The latter is in bijection with $H^2(F,K\_0)$, which is a reasonable well-understood object ($K\_0$ is a torus).


In particular:
1) If $G\_0$ is simply connected we get $G=G\_0\times F$ (only the direct extension)


2) Everything boils down to when $G\_0$ is a torus. If $S^1$ is the 1-circle, $Q$ is the group of quaternions of order 8 and if we define $G$ as the quotient of $S^1\times Q$ by the diagonal element of order 2, then we get a nontrivial extension (as we see by counting elements of order 2).


NB: you assume explicitly that $G$ is nilpotent. So I don't claim to classify all extensions of $G\_0$ by $F$.


