Local cross-sections for free actions of finite groups

Let $X=[-1,1]^\infty\setminus\{0\}$, which is a metrizable, locally compact space. Consider the two-element group $G$, and the free $G$-action on $X$ given by $(x_j)_{j=1}^\infty\mapsto (-x_j)_{j=1}^\infty$. We show that the fibration $X\to X/G$ has infinite Schwarz genus.

Consider the $n$-sphere $S^n$ with the antipodal $G$-action. Then $S^n$ can be embedded equivariantly into $X$ for all $n$. (Use an equivariant map $S^n\to [-1,1]^{n+1}\setminus\{0\}$.) By the Lusternik–Schnirelmann theorem (a strengthening of the Borsuk-Ulam theorem), $S^n$ cannot be covered by $n+1$ closed sets that do not contain antipodal points. It follows that the Schwarz genus of $S^n\to S^n/G$ is at least $n+2$. Since the Schwarz genus of $X\to X/G$ is an upper bound for the Schwarz genus of $S^n\to S^n/G$, it follows that $X\to X/G$ has infinite Schwarz genus.

There is a general cohomological lower bound for the Schwarz genus of a map $p:E\to B$. Namely, if there are cohomology classes $x_1,\ldots , x_k\in H^*(B)$ such that $0=p^*(x_i)\in H^*(E)$ for all $i=1,\ldots , k$ and $x_1\cup\cdots \cup x_k \neq 0$, then the genus of $p$ is greater than $k$. Here the coefficients are completely arbitrary, in particular can be twisted. (This is a generalisation of the cup-length lower bound for Lusternik-Schnirelmann category, since the LS-category of a space $X$ is equal to the genus of any fibration over $X$ with contractible total space.)

So you can get many counter-examples using this cohomological criterion. In fact, whenever $X$ is a contractible CW-complex then it is a model for $EG$, and $X/G$ is a model for $BG$. The cup-length of $BG$ is always infinite for a finite group $G$ (with appropriately chosen, possibly twisted coefficients). This generalises the example in Hannes Thiel's answer.