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Let $N$ be a connected nonorientable surface with or without boundary andpunctures, and $j\colon S\rightarrow N$ be the orientation double covering. Ithas previously been proved that the orientation double covering $j$ induces anembedding $\iota\colon\mathrm{Mod}(N)$ $\hookrightarrow$ $\mathrm{Mod}(S)$ withone 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 theembedding $\mathrm{Mod}(F') \hookrightarrow \mathrm{Mod}(F)$ induced by aninclusion of a pair of possibly nonorientable surfaces $F' \subset F$ is aquasi-isometric embedding.

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover