  We provide infinite families of quadratic number fields with everywhere
unramified Galois extensions of Galois group $SL_2(7)$ and $2.A_7$,
respectively. To my knowledge, these are the first instances of infinitely many
such realizations for perfect groups which are not generated by involutions, a
property which makes them difficult to approach for the problem in question and
leads to somewhat delicate local-global problems in inverse Galois theory.
