Centralizer of a specific permutation

The commutativity condition $\mu\pi=\pi\mu$ can be written as $\mu\pi\mu^{-1}=\pi$. Thus a permutation $\pi$ commutes with $\mu$ if and only if relabeling the cycle structure of $\pi$ according to $\mu$ leaves $\pi$ invariant. That means that for a given cycle of $\pi$, either the cycle has to have even length and pairs of elements at maximal distance from each other in the cycle must be mapped to each other by $\mu$, or the cycle has to be matched with another cycle of the same length and $\mu$ has to map the elements of the cycles to each other in order. (In particular, $\pi$ must have an even number of cycles of every odd length; note the similarity to the condition for a permutation to be the square of some permutation, which is that there must be an even number of cycles of every even length.)

There are $2^kk!$ such permutations, and they are in bijection with $\mathbb Z_2^k\times S_k$. The permutation in the centralizer of $\mu$ corresponding to $(v,\rho)\in\mathbb Z_2^k\times S_k$ is generated from the cycle structure of $\rho$ by making binary choices according to the elements of $v$. For each cycle $\gamma$ of $\rho$, the element $v_i$ corresponding to the least element $i$ in $\gamma$ determines whether $\gamma$ gets cloned to form a pair of cycles mapped to each other under relabeling with $\mu$, or whether $\gamma$ gets doubled in length to form a cycle mapped to itself under relabeling with $\mu$. The elements $v_j$ corresponding to the remaining elements $j\ne i$ in $\gamma$ determine whether $j$ or $n+1-j$ takes the place of $j$ in the cloned or doubled cycle(s).

The centralizer of $\mu$ is a example of a wreath product: Specifically, it is isomorphic to $C_{2} \wr S_{k},$ so its order is indeed $k! 2^{k}.$ The "base group" is elementary Abelian group of order $2^{k},$ and is generated by the $k$-disjoint transpositions appearing in $\mu,$ and the $S_{k}$ permutes these $k$ generating transpositions around as if they were individual points. But it should be noted that the centralizer is definitely not a direct product- it is a particular kind of semi-direct product, but the $S_{k}$ induces non-trivial automorphisms of the base group.