Hereditary cotorsion pairs on extriangulated subcategories

Let $\mathcal B$ be an extriangulated category with enough projectives and enough injectives. We define a proper $m$-term subcategory $\mathcal G$ on $\mathcal B$, which is an extriangulated subcategory. Then we give a correspondence between cotorsion pairs on $\mathcal G$, support $\tau$-tilting subcategories on an abelian quotient of $\mathcal G$ when $m=2$. If such $\mathcal G$ is induced by a hereditary cotorsion pair, then we give a correspondence between cotorsion pairs on $\mathcal G$ and intermediate cotorsion pairs on $\mathcal B$ under certain assumptions. At last, we study an important property of such extriangulated subcategory $\mathcal G$.