  Let $N$ be a connected nonorientable surface with or without boundary and
punctures, and $j\colon S\rightarrow N$ be the orientation double covering. It
has previously been proved that the orientation double covering $j$ induces an
embedding $\iota\colon\mathrm{Mod}(N)$ $\hookrightarrow$ $\mathrm{Mod}(S)$ with
one exception. In this paper, we prove that this injective homomorphism $\iota$
is a quasi-isometric embedding. The proof is based on the semihyperbolicity of
$\mathrm{Mod}(S)$, which has already been established. We also prove that the
embedding $\mathrm{Mod}(F') \hookrightarrow \mathrm{Mod}(F)$ induced by an
inclusion of a pair of possibly nonorientable surfaces $F' \subset F$ is a
quasi-isometric embedding.
