  Continuing the line of thought of an earlier work, we provide the first
infinite family of quadratic number fields with everywhere unramified Galois
extensions of Galois group $SL_2(5)$, the (unique) smallest nonsolvable group
for which this problem was previously open. Our approach also improves upon
previous work by yielding the first infinite family of real-quadratic fields
possessing an unramified Galois extension whose Galois group is perfect and not
generated by involutions. Our result also amounts to a new existence result on
quintic number fields with squarefree discriminant and additional local
conditions.
