  We study the sublattices of the rank $2^d$ Barnes-Wall lattices $\bw d$ which
occur as fixed points of involutions. They have ranks $2^{d-1}$ (for dirty
involutions) or $2^{d-1}\pm 2^{k-1}$ (for clean involutions), where $k$, the
defect, is an integer at most $\frac d 2$. We discuss the involutions on $\bw
d$ and determine the isometry groups of the fixed point sublattices for all
involutions of defect 1. Transitivity results for the Bolt-Room-Wall group on
isometry types of sublattices extend those in \cite{bwy}. Along the way, we
classify the orbits of $AGL(d,2)$ on the Reed-Muller codes $RM(2,d)$ and
describe {\it cubi sequences} for short codewords, which give them as Boolean
sums of codimension 2 affine subspaces.
