Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a maximal $(G)$ and minimal $(1)$ element. On the other hand, we can remove these offending maximal and minimal elements to get something interesting: define the reduced lattice $\Sigma'G$ which contains all the non-trivial proper subgroups of $G$ ordered by inclusion.
Define an operator $\mathcal{L}$ as follows $$\mathcal{L}G = \pi_1(\Sigma'G)$$ i.e., $\mathcal{L}$ maps each group $G$ to the fundamental group of $\Sigma'G$, choosing arbitrary basepoints in each connected component if necessary. Note that $\mathcal{L}$ is not an endo-functor on the category of groups since reduced lattices need not map to reduced lattices under a group homomorphism (a non-trivial kernel might map to $1$ for instance).
Which groups are fixed (up to isomorphism) by $\mathcal{L}$, i.e., satisfy $\mathcal{L}G \simeq G$?
There is tons of literature on subgroup lattices (including a paper which seemed relevant but turned out to only consider intervals in the lattice) so I'm not sure where to look for helpful results. I don't expect a complete solution, it will be nice to even just have some idea of what types of groups can arise as fundamental groups of reduced lattices.
