I'd like to get some clarity about how nice the problem of decomposing semigroups using the wreath product is compared to the problem of decomposing groups, and where this apparent difference in niceness comes from.
Looking at the Krohn-Rhodes theorem for finite semigroups, one can decompose a finite semigroup as a wreath product of finite simple groups and really basic "reset automata" semigroups.
If I understand the Krohn-Rhodes theorem correctly, if one applies it to a finite group, you don't get any "reset automata" semigroups, so it gives us a decomposition of our group as the wreath product of simple groups. Is this correct?
On the other hand, Meaning of factorization of groups and looking into the extension problem for finite groups leads me to doubt that the case for finite groups is that simple.
So is it not as simple as "there's some kind of wreath product-like construction that allows us to reconstruct a group from its factors"?
I'd greatly appreciate some clarity about this.
I am afraid that you missed an important point. Krohn-Rhodes theorem states that every finite semigroup divides a wreath product of finite simple groups and 2-element "reset automata" semigroups. Here "divides" means "quotient of a subsemigroup". In the group case, it is relatively easy to show that every finite group divides a wreath product of finite simple groups, but the proof of Krohn-Rhodes theorem is much more involved.