We classify "nice" loop envelopes to Bruck loops of 2-power exponent under
the assumption that every nonabelian simple section of $G$ is either passive or
isomorphic to $\PSL_2(q)$, $q-1 \ge 4$ a 2-power. The hypothesis is verified in
a separate paper. This paper is a continuation of the work by Aschbacher,
Kinyon and Phillips on finite Bruck loops [AKP]. In [BS3] we applied these
results and get a neat description of the structure of the finite Bruck loops.