Suppose that we know a discrete group acts on the geometric realization of a simplicial set. Is there some way to understand how the corresponding action works on the simplicial set?

For example, if we have the action of $\mathbb{Z}_2$ on the topological space $S^n$ given by $t\cdot x=-x$, where $t$ is the generator of $\mathbb{Z}_2$ and $x\in S^n$, then what is the corresponding action of $\mathbb{Z}_2$ on the simplicial set$S^n$.

The simplicial structure of $S^n$ is given by $S^n_k=${ $* $} for $k< n$ and $S^n_k=\lbrace *, (i_0,i_1,\dots, i_k)\;|\; 0\leq i_0\leq\cdots\leq i_k\leq n \rbrace$ for $k\geq n$. Faces and degeneracies are given by "deleting and doubling" for catching simplicial identities.

What is the action of $\mathbb{Z}_2$ on $S^n_k$ for each $k$? When $k< n$, the action is trivial. What about the case of $k\geq n$?

pleaseadd that information to the queston body itself (and then remove your answer, which is not an answer!) $\endgroup$ – Mariano Suárez-Álvarez Nov 8 '10 at 5:16