  A classical result of Conway and Pless is that a natural projection of the
fixed code of an automorphism of odd prime order of a self-dual binary linear
code is self-dual. In this paper we prove that the same holds for involutions
under some (quite strong) conditions on the codes. In order to prove it, we
introduce a new family of binary codes: the semi self-dual codes. A binary
self-orthogonal code is called semi self-dual if it contains the all-ones
vector and is of codimension 2 in its dual code. We prove upper bounds on the
dual distance of semi self-dual codes. As an application we get the following:
let C be an extremal self-dual binary linear code of length 24m and s in Aut(C)
be a fixed point free automorphism of order 2. If m is odd or if m=2k with
binom{5k-1}{k-1} odd then C is a free F_2<s>-module. This result has quite
strong consequences on the structure of the automorphism group of such codes.
