Centralizers in Hall's universal and direct limits of finitary symmetric groups

We study the centralizers of finite subgroups in Hall's universal group. We describe the structure of the centralizers of arbitrary finite subgroups in the groups S(\xi) and FSym(\kappa)(\xi) where S(\xi)'s are obtained as direct limits of finite symmetric groups and FSym(\kappa)(\xi)'s are obtained as direct limits of finitary symmetric groups on the set of infinite cardinality \kappa.