Here we obtain an extension of the above computability result for free $G$-spaces and equivariant maps. The input $G$-spaces $X$ and $Y$ can be given as finite *simplicial sets* (generalizations of finite simplicial complexes, see [@Friedm08]), and the free action of $G$ is supposed to be simplicial. The simplicial sets and the $G$-actions on them are described by a finite table.

\[theorem:equivariant\] Let $G$ be a finite group. There is an algorithm that, given finite simplicial sets $X$ and $Y$ with free simplicial actions of $G$, such that $Y$ is $d$-connected, $d\geq 1$, and $\dim X\leq 2d+1$, decides the existence of a continuous equivariant map $X\to Y$.

If such a map exists and $\dim X\leq 2d$, then the set $[X,Y]$ of all equivariant homotopy classes of equivariant continuous maps can be equipped with the structure of a finitely generated abelian group, and the algorithm outputs the isomorphism type of this group.

For fixed $G$ and ${{d}}$, this algorithm runs in polynomial time.

Here the isomorphism type is output as an abstract abelian group given by a (finite) number of generators and relations. Furthermore, there is an algorithm that, given an equivariant simplicial map $\ell{\colon}X\to Y$, computes the element of this 