by the dashed arrow, that makes both triangles commute. We call such an $\ell$ a *solution* of the lifting-extension problem .

Moreover, if such an $\ell$ exists, we would like to compute the structure of the set $[X,Y]^A_B$ of all solutions, up to (equivariant) *fibrewise* homotopy *relative to $A$*.[^3] More concretely, in the cases covered by our algorithmic results, we will be able to equip $[X,Y]^A_B$ with a structure of an abelian group, and the algorithm computes the isomorphism type of this group.

The generalized lifting-extension problem {#the-generalized-lifting-extension-problem .unnumbered}
-----------------------------------------

Spaces appearing in a fibration $\varphi {\colon}Y{\mathbin{\:\!\!\xymatrix@1@C=15pt{{}\ar@{->>}[r]^{} & {}}}}B$ must typically be represented by infinite simplicial sets[^4], and their representation as inputs to an algorithm can be problematic. For this reason, we will consider a *generalized lifting-extension problem*, where, compared to the above, $\varphi{\colon}Y\to B$ can be an arbitrary equivariant map, *not* necessarily a fibration.

In this case, it makes no sense from the homotopy point of view to define a solution as a map $X\to Y$ making both triangles commutative. A homotopically correct way of defining a solution is to consider pairs $(\ell,h)$, where $\ell{\colon}X\to Y$ is a map for which the upper triangle commutes strictly and the lower 