  The $p$-group generation algorithm from computational group theory is used to
obtain information about large quotients of the pro-2 group $G = \text{Gal}
(k^{nr,2}/k)$ for $k = \mathbb{Q}(\sqrt{d})$ with $d = -445, -1015, -1595,
-2379$. In each case we are able to narrow the identity of $G$ down to one of a
finite number of explicitly given finite groups. From this follow several
results regarding the corresponding 2-class tower.
  This is a revised version of ANT-0302.
