Counterexamples to a conjecture of Lemmermeyer
  We produce infinitely many finite 2-groups that do not embed with index 2 in any group generated by involutions. This disproves a conjecture of Lemmermeyer and restricts the possible Galois groups of unramified 2-extensions, Galois over the rationals, of quadratic number fields. 
Nigel Boston, Charles Leedham-Green