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
$ {\BBF }_{2}\langle g \rangle $
-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.