In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is mentioned:

Theorem 1. $H$-limits commute with $G$-colimits in $\mathsf{Set}$ iff no nontrivial quotient of $H$ is isomorphic to a subquotient of $G$.

On the nlab page about commutativity of limits and colimits, it's also mentioned that taking orbits under the action of a finite group commutes with cofiltered limits, i.e:

Theorem 2. Let $G$ be a finite group and $\mathsf C$ be a small cofiltered category. Let $F:\mathsf C\longrightarrow G\mathsf{Set}$ be a functor. Then $(\varprojlim F)/G\cong \varprojlim _{j\in F}(F(f)/G)$.

I'm reasonably familiar with category theory, but I've only ever taken one course on groups - a first semester course about finite groups. I'm curious whether one could derive from these commutation results some basic facts in the theory of (finite) groups.

What kind of familiar elementary facts about (finite) groups can be derived from these results?

Update: This question has now been crossposted to MO.


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