This is a continuation of this [question][1], where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}_2$ of the $n$-cycle. We needed to know the explicit structure of its centralizer in $\mathrm{GL}_{n}(\mathbb{F}_2)$ (more precisely, of its factor $Z(n)$ over the subgroup generated by $C$), and we computed it. It's an abelian group, and so we were able to compute its decomposition into the sum of cyclic subgroups, as follows, where $n=2^km$, with $m$ odd. 
$$
Z(n)=\bigoplus_{i=2}^k (\mathbb{Z}/\mathbb{Z}_{2^{k+1-i}})^{2^{i-2}m}\oplus \mathbb{Z}/\mathbb{Z}_{2^{k}}^{m-1}\oplus Z(m).
$$
The decomposition of $Z(m)$ is also explicit, and can be described in terms of factors of the $\mathbb{F}_2$-irreducible factors of the polynomial $x^m+1\in\mathbb{F}_2[x]$. 

We wonder if this was already done.  A comment to [loc.cit.][1] says that the case $n=2^k$ was done in in Prop. XI(5.7) of Bass's "Algebraic K-Theory". It is useful to note that the question is equivalent to describing the group of units in the $\mathbb{F}_2$-group algebra of $\mathbb{Z}/n\mathbb{Z}$. It appears that in general the structure of the group of units in the group algebra of $\mathbb{Z}/n\mathbb{Z}$ over a finite field is not known, as one can find fresh publications settling particular cases, cf. e.g. [Int.J. Group Theory v.2, No. 4 (2013), 1-6][3].


  [1]: https://mathoverflow.net/questions/138025/centralizer%20of%20the%20order%202%5Ek%20cyclic%20permutation%20matrix%20over%20F_2
  [3]: http://www.theoryofgroups.ir/?_action=articleInfo&article=2643&vol=252