  We construct a class $\Theta_{\mathscr{R}}$ of homomorphisms from a Specht
module $S_{\mathbb{Z}}^{\lambda}$ to a signed permutation module
$M_{\mathbb{Z}}(\alpha|\beta)$ which generalises James's construction of
homomorphisms whose codomain is a Young permutation module. We show that any
$\phi \in
\operatorname{Hom}_{{\mathbb{Z}}\mathfrak{S}_{n}}\big(S_{\mathbb{Z}}^\lambda,
M_{\mathbb{Z}}(\alpha|\beta)\big)$ lies in the $\mathbb{Q}$-span of
$\Theta_{\text{sstd}}$, a subset of $\Theta_{\mathscr{R}}$ corresponding to
semistandard $\lambda$-tableaux of type $(\alpha|\beta)$. We also study the
conditions for which $\Theta^{\mathbb{F}}_{\mathrm{sstd}}$ - a subset of
$\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^\lambda,M_{\mathbb{F}}(\alpha|\beta)\big)$
induced by $\Theta_{\mathrm{sstd}}$ - is linearly independent, and show that it
is a basis for
$\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^\lambda,M_{\mathbb{F}}(\alpha|\beta)\big)$
when $\mathbb{F}\mathfrak{S}_{n}$ is semisimple.
