the main result of this paper.

\[thm:main\_theorem\] Let $G$ be a finite group and let an instance of the generalized lifting-extension problem be input as follows: $A$, $X$, $Y$, $B$ are finite simplicial sets with free simplicial actions of $G$, $A$ is an (equivariant) simplicial subset of $X$, and $f$, $g$, $\varphi$ are equivariant simplicial maps. Furthermore, both $B$ and $Y$ are supposed to be $1$-connected, and the *homotopy fibre*[^7] of $\varphi{\colon}Y\to B$ is supposed to be $d$-connected for some $d\geq 1$.

There is an algorithm that, for $\dim X\leq 2d+1$, decides the existence of a solution. Moreover, if $\dim X\leq 2d$ and a solution exists, then the set $[X,Y]_B^A$ can be equipped with the structure of an abelian group, and the algorithm computes its isomorphism type. The running time of this algorithm is polynomial when $G$ and ${{d}}$ are fixed.

As in Theorem \[theorem:equivariant\], the isomorphism type means an abstract abelian group (given by generators and relations) isomorphic to $[X,Y]^A_B$. Given an arbitrary diagonal $\ell{\colon}X\to Y$ in the considered square, one can compute the element of this group that the solution $j\ell{\colon}X\to Y'$ represents.

Constructing the abelian group structure on $[X,Y]_B^A$ will be one of our main objectives. In 