This question has a somewhat similar flavor to mathoverflow.net/questions/153433/…

My favorite example is the cross polytope (i.e., generalized octahedron), which is the independence complex of the disjoint union of $K_2$'s.

Matoušek's nice book "Using the Borsuk-Ulam theorem" should be mentioned here. It is a grad-student accessible, book-length treatment of the Lovász solution to the Kneser problem and more recent improvements (and all the background you need to understand them).

@user47659: Yes, that's what I mean. It's easiest to prove if you focus on the (minimal) non-faces of $\Delta$ and the circuits of matroids, rather than the faces and independent sets. Here face == independent set, in different terminology.

Yes, StructureDescription is limited. But is there a better alternative for getting a human-readable description of groups, suitable for MathOverflow? (I guess one easy improvement would be to only ask for the structure of the outer automorphism group.)

It's not a complete answer to your question, but if you want to get some intuition, GAP (and hence SAGE) has a library of all groups of order $p^4$ built in. See gap-system.org/Packages/sgl.html

Maybe the fact that Gaschütz's Theorem (see my answer below) doesn't hold when $C$ is not abelian will give a counterexample in this case? (But I didn't immediately find the counterexample, even without the cyclic constraints.)

By the way, you can find (a slightly more general version of) Gaschütz's Theorem as Theorem 3.3.2 of "The theory of finite groups" by Kurzweil and Stellmacher. The result is very useful in situations like this question, and in my opinion it should be more widely known.

Of course, if you do the same with a dodecahedron with the correct twist, you famously get the Poincare homology sphere! en.wikipedia.org/wiki/…

The relevant page of Hilton's text can be reached (at least for now) by Google Books at books.google.com/…

If I'm understanding Hilton's formulation, I think it should suffice to take one such disc, corresponding with $f^{-1}$ of a single point. Of course, if $\Delta$ is sufficiently finely subdivided, it shouldn't matter what color you pick.

No, I don't think maximal is necessary. Indeed, you might get a bit of extra power by looking also at meet-irreducible (but not necessarily maximal) subgroups.