On the Galois groups of the 2-class towers of some imaginary quadratic fields

Let and knr,2 be the maximal unramified 2-extension of k. To show that knr,2/k is finite, Michael Bush gave 8 possible presentations of finite groups for G=Gal(knr,2/k). However, his methods did not further isolate G. We eliminate 4 of the possibilities, and explain how to isolate G, although carrying out the latter strategy is beyond current technological capabilities. We also discuss related examples.