Let $C$ be the $2^k\times 2^k$-permutation matrix over $\mathbb{F}_2$ of the $2^k$-cycle. We needed to know the structure of its centralizer in $\mathrm{GL}_{2^k}(\mathbb{F}_2)$, and we computed it - it was not too easy. It's an abelian group, and so we were able to compute the decomposition of the quotient of the centralizer over the subgroup generated by $C$ into the sum of cyclic subgroups, as follows. $$ \bigoplus_{i=2}^k (\mathbb{Z}/\mathbb{Z}_{2^{k+1-i}})^{2^{i-2}}. $$

We wonder if this was already done. (We also did this for more general case of other primes, not only 2, formulas are similar).

Update: see centralizer of a n-cyclic permutation matrix over F_2 in GL(n,2) for a follow-up question.