for $\varphi{\colon}Y\to B$; see Section \[s:main\_proofs\]. Moreover, $Y'$ is not determined uniquely, and thus neither are the solutions of the generalized lifting-extension problem. However, rather standard considerations show that the *existence* of a solution and the *isomorphism type* of $[X,Y']_B^A$ as an abelian group are independent of the choice of $Y'$.

Examples of lifting-extension problems {#examples-of-lifting-extension-problems .unnumbered}
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In order to understand the meaning of the (generalized) lifting-extension problem, it is instructive to consider some special cases.

1.  (Classification of extensions.) First, let us consider $G=\{e\}$ trivial (thus, the equivariance conditions are vacuous) and $B$ a point (which makes the lower triangle in the lifting-extension problem superfluous). Then we have an *extension problem*, asking for the existence of a map $\ell{\colon}X\to Y$ extending a given $f{\colon}A\to Y$. Moreover, $[X,Y]$ is the set of appropriate homotopy classes of such extensions.[^5] We remind that this problem is undecidable when $\dim X$ is not bounded, according to [@ext-hard].

2.  (Equivariant maps.) Here $G$ is nontrivial, $A=\emptyset$, and $B=EG$, a contractible free $G$-space (it is unique up to equivariant homotopy equivalence). For every free $G$-space $Z$, there is an equivariant map $c_Z{\colon}Z\to EG$, unique up to equivariant homotopy. If we set $g=c_X$ and $\varphi=c_Y$ 