The main result of this paper is the analogue for topological groups of a theorem due to S. Dierolf which states that there exists a nonlocally convex twisted sum of the locally convex spaces

$X$ and

$Y$ if and only if there exists a nonlocally convex twisted sum of

$X$ and

$\mathbb{R}$. It reads as follows: Given two abelian locally quasi-convex groups

$H$ and

$G$ there exists a nonlocally quasi-convex extension of

$H$ and

$G$ if and only if there exists a nonlocally quasi-convex extension of the circle group

${S}^{1}$ and

$G$. The proof relies on methods of homological algebra.