The multiplicative Jordan decomposition in group rings, II

We classify the finite 2-groups G whose integral group rings Z[G] have the multiplicative Jordan decomposition property. In addition to those cases already known, these include three further cases of order thirty-two and no others. We also give a theorem which severely restricts the structure of those finite groups G with Z[G] having this property which are not of order a2b3 for some a,b.