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

Let $N$ be a connected nonorientable surface with or without boundary and punctures, and $j\text{\hspace{0.17em}⁣}:S\to N$ be the orientation double covering. It has previously been proved that the orientation double covering $j$ induces an embedding $\iota \text{\hspace{0.17em}⁣}:\mathrm{M}\mathrm{o}\mathrm{d}(N)$ $\hookrightarrow$ $\mathrm{M}\mathrm{o}\mathrm{d}(S)$ 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 $\mathrm{M}\mathrm{o}\mathrm{d}(S)$, which has already been established. We also prove that the embedding $\mathrm{M}\mathrm{o}\mathrm{d}(F^{\mathrm{\prime }})\hookrightarrow \mathrm{M}\mathrm{o}\mathrm{d}(F)$ induced by an inclusion of a pair of possibly nonorientable surfaces $F^{\mathrm{\prime }}\subset F$ is a quasi-isometric embedding.