2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77341We 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.Number Theorytext