Is there an abelian category of topological groups? There are lots of reasons why the category of topological abelian groups (i.e. internal abelian groups in $\bf Top$) is not an abelian category. So I'm wondering:

Is there a "suitably well behaved" subcategory of $\bf Top$, say $\bf T$, such that $\bf Ab(T)$ is an abelian category? 

My first guess was to look for well behaved topological spaces (locally compact Hausdorff, compactly generated Hausdorff, and so on...) Googling a little shows me that compactly generated topological groups are well known animals, but the web seems to lack of a more categorical point of view.
Any clue? Thanks in advance.

Perhaps it's worth posting an update here: by replacing topological spaces by condensed sets resp. pyknotic sets we can get an abelian category of abelian group objects, the condensed abelian groups resp. the pyknotic abelian groups; for a discussion of the first see Scholze's Lectures on Condensed Mathematics and for a discussion of the second see Barwick-Haine's Pyknotic objects, I. Basic notions.
If I'm reading correctly, the category of compactly generated topological spaces embeds fully faithfully as a reflective subcategory of either condensed or pyknotic sets, and I believe this implies that the category of compactly generated abelian groups (not necessarily Hausdorff!) embeds fully faithfully as a reflective subcategory of either condensed or pyknotic abelian groups. If that's right, then this embedding preserves limits but it does not preserve colimits, and this is necessary to get an abelian category since we need to alter the behavior of cokernels, e.g. the cokernel of the map from $\mathbb{R}$ with the discrete topology to $\mathbb{R}$ with the Euclidean topology is nontrivial, which Scholze memorably uses as a motivating example.
