the paper {#plan-of-the-paper .unnumbered}
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In the second section, we give an overview of equivariant effective homological algebra that we use in the rest of the paper. The third section is devoted to the algorithmic construction of an equivariant Moore–Postnikov tower. The proofs of Theorems \[theorem:equivariant\] and \[thm:main\_theorem\], without their polynomial time claims, are given in the following section, although proofs of its two important ingredients are postponed to Sections $5$ and $6$. In the fifth section, we construct a certain weakening of an (equivariant and fibrewise) [H-]{}space structure on stable stages of Moore–Postnikov towers with sections. In the sixth section, we show how this structure enables one to endow the sets of homotopy classes with addition in an algorithmic way. Finally, we derive an exact sequence relating $[X,{{P_{{n}}}}]^A_B$ to $[X,{{P_{{{n}}-1}}}]^A_B$ and thus enabling an inductive computation. In the seventh section, we provide proofs that we feel would not fit the previous sections. In the last section, we prove polynomial bounds for the running time of our algorithms.

Equivariant effective homological algebra {#sec:equi_eff_hlgy_alg}
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Basic setup
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For a simplicial set, the face operators are denoted by $d_{{i}}$, and the degeneracy operators by $s_{{i}}$. The standard ${{m}}$-simplex ${\Delta^{{{m}}}}$ is a 