  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 $\mathcal{M}
(N)$ be the mapping class group of $N$. We show that, if $\mathcal{G}$ is a
finite index subgroup of $\mathcal{M} (N)$ and $\varphi: \mathcal{G} \to
\mathcal{M} (N)$ is an injective homomorphism, then there exists $f_0 \in
\mathcal{M} (N)$ such that $\varphi (g) = f_0 g f_0^{-1}$ for all $g \in
\mathcal{G}$. We deduce that the abstract commensurator of $\mathcal{M} (N)$
coincides with $\mathcal{M} (N)$.
