Let $K$ and $Q$ be discrete groups, with $K$ abelian. Central extensions of $Q$ by $K$, i.e. short exact sequences $1 \to K \to G \to Q \to 1$ such that $K$ is lies in the center of $G$, are classified (up to equivalence) by $H^2(Q;K)$, where $K$ is a trivial $Q$-module. There are also corresponding versions of this statement for (a) non-central extensions, and (b) $K$ not necessarily abelian.
My question is whether there exists a way of classifying compact Lie groups which arise as extensions of $Q$ by $K$ when $K$ is assumed to be a compact and connected Lie group. I would be happy to know what happens even in special cases, e.g. when $K$ is circle.
If you have a central extension $1\rightarrow S^1\rightarrow G\rightarrow Q\rightarrow 1$ where $Q$ is compact and suppose that you have local continuous sections, you can find a cover $(U_i)$ of $Q$ with sections $s_i:U_i\rightarrow G$, $s_{ij}:U_i\cap U_j\rightarrow S^1$ defined by $s_{ij}=s_is_j^{-1}$ is a Cech cocycle defined on $G$ which takes it values in $S^1$, you have a classifying cocycle $s$ defined by $s_{ij}$ whose class $[s]\in H^1(Q,S^1)=H^2(Q,\mathbb{Z})$ in fact it is the Chern class of a complex line bundle defined on $Q$.