On the group cohomology of groups of the form $Z^n Z/m$ with $m$ free of squares

We provide an explicit computation of the cohomology groups (with untwisted coefficients) of semidirect products of the form Zⁿ Z/m with m free of squares, by means of formulas that only depend on n, m and the action of Z/m on Zⁿ. We want to highlight the fact that we are not impossing any conditions on the Z/m-action on Zⁿ, and as far as we know our formulas are the first in the literature in this generality. This generalizes previous computations of L\"uck-Davis and Adem-Ge-Pan-Petrosyan. In order to show that our formulas are usable, we develop a concrete example of the form Z⁵ Z/6 where its cohomology groups are described in full detail.