On Olga Taussky's article "A Remark on the class field tower" there is a note on the last page where she states that P. Hall pointed out to her that exists 3 different groups of order $2^k$ for k > 3 with abelianization of type (2,2). But there is no reference to some work of Hall or other author about it. It states also that the study of such groups can be reduced to the case of groups of order 16.
She states that this result can be derived from the present work, but Im not familiarized with class field tower at all, and want an approach more related to classical group theory. I was able to find out my way for groups of order 16 using their classification.
My question is: How to prove that exists 3 different groups of order $2^k$ for k > 3 with abelianization of type (2,2), and that its study can be reduced to the case where the group has order 16?