Injective homomorphisms of mapping class groups of non-orientable surfaces

Let $N$ be a compact, connected, non-orientable surface of genus $\rho$ with $n$ boundary components, with $\rho \ge 5$ and $n\ge 0$, and let $$\mathcal{M}$ (N)$bethemappingclassgroupof$N$\mathrm{.}Weshowthat,if$$\mathcal{G}$$ is a finite index subgroup of $\mathcal{M}(N)$ and $\varphi: $\mathcal{G}$ \to $\mathcal{M}$ (N)$isaninjectivehomomorphism,thenthereexists$f_0 \in $\mathcal{M}$ (N)$suchthat$\varphi (g) = f_0 g f_0^{-1}$forall$g \in $\mathcal{G}$$\mathrm{.}Wededucethattheabstractcommensuratorof$$\mathcal{M}$ (N)$coincideswith$$\mathcal{M}$ (N)$.