Hi all, nice to be here

I,ve been given this problem and have made some progress with it but am still slightly confused over the statement of it:

Essentially I need to find a simplicial complex (Simplicial complex - Wikipedia, the free encyclopedia) such that the alternating sum of the characters for the homology groups \sum (-1)^i h_i, where h_i is a character for the homology group H_i (Simplicial homology - Wikipedia, the free encyclopedia, better explained here) of the characters gives the character
  \chi (g)= |\{ (x,y) \in G | g=xyx^{-1}y^{-1}\}|

To explain in more detail:

is there a natural looking way to write this character as a difference of two permutation
characters for some classes of groups? This would follow from the
simplicial complex statement by taking the permutation characters on
even/odd dimensional simplices and taking the difference of those

I think I have stated the problem correctly, but please ask if you think I am unclear as I don't quite understand it. My main issues are:

Surely you cannot make a simplicial complex with homology groups isomorphic to any group? If not how to the characters on these groups relate to other groups like  S_n or SL(2,q) for example.

As regards to the statement on permutation characters, how can you have a permutation character on a homology group, when the group is not acting on anything? Is there a natural way to see this as acting on the simplicial complex?

Sorry for the long post, If you think you may be able to help with even just a tiny part of my question it would be a great help.

Many thanks,