  The automorphism group of a binary doubly-even self-dual code is always
contained in the alternating group. On the other hand, given a permutation
group $G$ of degree $n$ there exists a doubly-even self-dual $G$-invariant code
if and only if $n$ is a multiple of 8, every simple self-dual $\F_2G$-module
occurs with even multiplicity in $\F_2^n$, and $G$ is contained in the
alternating group.
