  We present an algorithm that, given finite simplicial sets $X$, $A$, $Y$ with
an action of a finite group $G$, computes the set $[X,Y]^A_G$ of homotopy
classes of equivariant maps $\ell \colon X \to Y$ extending a given equivariant
map $f \colon A \to Y$ under the stability assumption $\dim X^H \leq 2
\operatorname{conn} Y^H$ and $\operatorname{conn} Y^H \geq 1$, for all
subgroups $H\leq G$. For fixed $n = \operatorname{dim} X$, the algorithm runs
in polynomial time. When the stability condition is dropped, the problem is
undecidable already in the non-equivariant setting.
  The algorithm is obtained as a special case of a more general result: For
finite diagrams of simplicial sets $X$, $A$, $Y$, i.e. functors
$\mathcal{I}^\mathrm{op} \to \mathsf{sSet}$, in the stable range
$\operatorname{dim} X \leq 2 \operatorname{conn} Y$ and $\operatorname{conn} Y
> 1$, we give an algorithm that computes the set $[X, Y]^A$ of homotopy classes
of maps of diagrams $\ell \colon X \to Y$ extending a given $f \colon A \to Y$.
Again, for fixed $n = \dim X$, the running time of the algorithm is polynomial.
  The algorithm can be utilized to compute homotopy invariants in the
equivariant setting -- for example, one can algorithmically compute equivariant
stable homotopy groups. Further, one can apply the result to solve problems
from computational topology, which we showcase on the following Tverberg-type
problem: Given a $k$-dimensional simplicial complex $K$, is there a map $K \to
\mathbb{R}^{d}$ without $r$-tuple intersection points? In the metastable range
of dimensions, $rd \geq (r+1)k +3$, the result of Mabillard and Wagner shows
this problem equivalent to the existence of a particular equivariant map. In
this range, our algorithm is applicable and, thus, the $r$-Tverberg problem is
algorithmically decidable (in polynomial time when $k$, $d$ and $r$ are fixed).
