If this question doesn't make sense or is otherwise poor quality, then I'm sorry.
Motivation:
As part of my research, I study virtually solvable (1) groups. These are goups that have a solvable subgroup of finite index.
I am interested in looking at this group theoretic property from a wider perspective to see if I can get inspiration for proving various things about virtually solvable groups.
The Question:
What would be a categorification of the property of being (virtually) solvable for groups?
Thoughts:
I think that if I have a categorification of solvable, then a categorification of virtually solvable would follow by using things like subobjects. Whether or not I could form such a definition would depend on the nature of the first categorification.
A good place to start might be group extensions, which, as far as I understand, are categorical; they're essentially short exact sequences. In fact, I think the latter (with a few bells & whistles) are pretty much a categorification of "solvable" anyway but I'm not sure. My understanding of these is minimal though. I would appreciate help with them.
Some Extra Context:
I'm completely self-taught when it comes to category theory. However, I have asked some questions of modest difficulty about category theory here on MSE in the past and the answers seem to make sense to me.
Please help :)
(1) See the "Related Concepts" section of the article linked to.