In this article we continue the investigation of the length of the narrow 2-class field tower of real quadratic number fields \(\textrm{k}\) whose discriminants are not a sum of two squares and for which their 2-class groups are elementary of order 4. Letting \(\textrm{G}\) equal the Galois group of the second Hilbert narrow 2-class field over \(\textrm{k}\) , and \([\textrm{G}_i]\) denote the lower central series of \(\textrm{G}\) , we give heuristic evidence that the length of the narrow 2-class field tower of \(\textrm{k}\) is equal to 2 when \(\textrm{G}/\textrm{G}_3\) is of type 64.150 (in the tables of Hall and Senior). We also give the formulation of the relevant unit groups of the narrow Hilbert 2-class field for these fields.
