Let \(e: 1\to N\to G\to K\to 1\) be an extension of a finite cyclic group \(N\) by a finite cyclic group \(K\). Then the literature already enables us to calculate \(H^*(G,\mathbb{Z})\) in principle. Step 1 is to reduce the calculation to a \(p\)-Sylow subgroup -- which will be either cyclic, abelian of rank 2 or non-abelian metacyclic. Step 2 is to handle each of these cases; the first two are well-known, for the third one can use the theory of characteristic classes to find a lower bound, an explicit resolution of \(\mathbb{Z}\) over \(\mathbb{Z} G\) to construct any additional generators, and the extension spectral sequence to put everything together. Step 3 is then to determine the stable elements for each prime \(p\) dividing \(| G|\). However although this procedure will certainly give the answer for any prescribed metacyclic group, its step- wise nature may obscure how the numerical relations defining the group in the case of composite order are reflected in cohomology. The author's aim in the present paper is to carry out a variant of the argument sketched above for the group as a whole, thus highlighting the role of the number theory. The details are extremely complicated, but are summarized in Theorem 0.3 of the introduction.