there are several new ingredients which we had to develop in order to make the computation possible. We will briefly describe these after the outline of the proof.

Our first tool is a Moore–Postnikov tower ${{P_{{n}}}}$ for $\varphi {\colon}Y{\rightarrow}B$ within the framework of (equivariant) effective algebraic topology (essentially, this means that all objects are representable in a computer); it is enough to construct the first $\dim X$ stages. It can be shown that $[X,Y]^A_B\cong[X,{{P_{{n}}}}]^A_B$ for ${{n}}\geq \dim X$ and so it suffices to compute inductively $[X,{{P_{{n}}}}]^A_B$ from $[X,{{P_{{{n}}-1}}}]^A_B$ for ${{n}}\leq \dim X$. This is the kind of problems considered in obstruction theory. Namely, there is a natural map $[X,{{P_{{n}}}}]^A_B\to[X,{{P_{{{n}}-1}}}]^A_B$ and it is possible to describe all preimages of any given homotopy class $[\ell] \in [X,{{P_{{{n}}-1}}}]^A_B$ using, in addition, an inductive computation of $[{\Delta^{1}}\times X,{{P_{{{n}}-1}}}]^{(\partial{\Delta^{1}}\times X)\cup({\Delta^{1}}\times A)}_B$. In general however, $[X,{{P_{{{n}}-1}}}]^A_B$ is infinite and it is thus impossible to compute $[X,{{P_{{n}}}}]^A_B$ as a union of preimages of all possible homotopy classes $[\ell]$ (on the other hand, if these sets are finite, the above description provides an algorithm, probably not very efficient, see [@Brown]).

For this reason, we use in the paper to a great advantage our second tool, an abelian group 