Isometric embedding of Teichmüller spaces

Question

Let $S_{g,k}$ be a genus $g$ surface with $k$ punctures. Let $\mathrm{Mod}(S)$ be the extended mapping class group of a surface, defined as the isotopy classes of the self-homeomorphisms of surface $S$. Let $\mathcal{T}(S_{g,k})$ be the Teichmüller space of $S$. Suppose there is an injective map $\phi : \mathrm{Mod}(S_{g,k}) \to \mathrm{Mod}(S_{g',k'})$, then how do I show that there is a continuous $\phi$-equivariant map $\Phi : \mathcal{T}(S_{g,k}) \to \mathcal{T}(S_{g',k'})$? Furthermore, if there is a surjective homomorphism from $\mathrm{Mod}(S_{g',k'})$ to $\mathrm{Mod}(S_{g,k})$, then how does one show that $\Phi$ is an isometric embedding with respect to Teichmüller metric? I have studied mapping class groups. I also know some important results about Teichmüller space, but I don't have a thorough understanding of Teichmüller space. I am currently studying the 2018 paper "Injections of mapping class groups" by Aramayona, Leininger and Souto (arXiv: https://arxiv.org/abs/0811.0841).