Ljusternik–Schnirelman theorem and $\beta f$

A generalization of the Aarts–Fokkink–Vermeer theorem ($k=1$ and the space is metrizable) is obtained. For every $k$ free homeomorphisms of an $n$-dimensional paracompact space onto itself, the coloring number is not greater than $n+2k+1$. As an application, it is obtained that for the free action of a finite group $G$ on a normal (finite dimensional paracompact) space $X$, the coloring number $LS$ and the genus $K$ of the space are related by
$$ LS(X;G)=K(X;G)+|G|-1\ \ (\leqslant\dim X+|G|). $$
As a corollary we prove that for all numbers $n$ and $k$ and the free action of the group $G=\mathbb Z_{2k+1}$ on the space $G*G*\cdots*G$ the coloring number is equal to $n+2k+1$ in the theorem formulated above. It is shown that for any $k$ pairwise permutable free continuous maps of an $n$-dimensional compact space $X$ into itself, the coloring number does not exceed $n+2k+1$. We generalise one theorem proved by Steinlein (about a free periodic homeomorphism), who gave a negative solution to Lusternik's problem. For any free map of a compact space into itself, the coloring number does not exceed the Hopf number multiplied by four.