The embedding of a cyclic permutable subgroup in a finite group

In earlier work, the authors described the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group. As might be expected, the even order case is considerably more complicated and we have found it necessary to divide it into two parts. This part deals with the situation where we have a finite group G with a cyclic permutable subgroup A satisfying the additional hypothesis that X is permutable in A₂X for all cyclic subgroups X of G (where A₂ is the 2-component of A).