Compactness in a short exact sequence of topological groups

Question

Suppose $H,G,K$ are abelian Hausdorff topological groups and $0\to H\overset{\alpha}\to G\overset{\beta}\to K\to 0$ an exact sequence of continuous homomorphisms.
If $H$ and $K$ are compact, can we conclude that $G$ is compact too?
(I know that the answer is yes if $\beta$ is open or closed, but can we drop this hypothesis?)

Answer 1

What about $H=0$, $G=S^1$ with the discrete topology, $K=S^1$ with the regular topology and $\beta$ the identity?