Suppose we are given group $G=\langle a_1,\ldots,a_n \mid R_1=1,\ldots R_m=1 \rangle$. Is there an algorithm which computes a finite presentation for the Schur multiplier, i.e. second homology group $K=H^2(G,\mathbb{Z})$? Can one at least solve a decision problem, whether $K$ is trivial or not?
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It is proved in "Some embedding theorems and undecidability questions for groups" by C. Gordon, that there is no algorithm that computes a presentation for the Schur multiplier from a presentation of $G$, and moreover there is no algorithm for the deciding whether the Schur multiplier is trivial or not. Some other similar related results are proved, as well. 

