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A classical result of Conway and Pless is that a natural projection of thefixed code of an automorphism of odd prime order of a self-dual binary linearcode is self-dual. In this paper we prove that the same holds for involutionsunder some (quite strong) conditions on the codes. In order to prove it, weintroduce a new family of binary codes: the semi self-dual codes. A binaryself-orthogonal code is called semi self-dual if it contains the all-onesvector and is of codimension 2 in its dual code. We prove upper bounds on thedual 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 withbinom{5k-1}{k-1} odd then C is a free F_2<s>-module. This result has quitestrong consequences on the structure of the automorphism group of such codes.

On involutions in extremal self-dual codes and the dual distance of semi self-dual codes