We call G a HallX -group if there exists a normal nilpotent subgroup N of G for which G=N 0 is an X -group. We call G a T0-group provided G=( G) is a T -group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally dene Hall X -groups and T0-groups where X 2 fT ;PT ;PST g; the classes PT and PST denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations. 2010 Mathematics Subject Classication: Primary: 20D10, 20D20, 20D35.