Automorphisms of Order $2p$ in Binary Self-Dual Extremal Codes of Length a Multiple of 24

Let C be a binary self-dual code with an automorphism g of order 2p, where p is an odd prime, such that g^p is a fixed point free involution. If C is extremal of length a multiple of 24, all the involutions are fixed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code C with coding theoretical ones of the subcode C(g^p) which consists of the set of fixed points of g^p, we prove that C is a projective F₂〈g 〉-module if and only if a natural projection of C(g^p) is a self-dual code. We then discuss easy-to-handle criteria to decide if C is projective or not. As an application, we consider in the last part extremal self-dual codes of length 120, proving that their automorphism group does not contain elements of order 38 and 58.