Centraliser of a fixed-point-free involution ($C_{S_{2m}}((12)\cdots (2m-12m))\cong C_2{\wr }_{\{1,\dots ,m\}}{S}_m$)

Commutation with a fixed-point-free involution preserves its partner relation. A commuting permutation can permute the $m$ pairs and independently swap each pair. This gives the permutation wreath product with order $2^{m}m!$.