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In this work we introduce the notion of higher $\mathbb{E}$-extension groupsfor an extriangulated category $\mathcal{C}$ and study the quotients$\mathcal{X}_{n+1}^{\vee}/[\mathcal{X}]$ and$\mathcal{X}_{n+1}^{\wedge}/[\mathcal{X}]$ when $\mathcal{X}$ is an$(n+2)$-rigid subcategory of $\mathcal{C}$. We also prove (under mildconditions) that each one is equivalent to a suitable subcategory of thecategory of functors of the stable category of $\mathcal{X}_{n}^{\vee}$ and theco-stable category of $\mathcal{X}_{n}^{\wedge}$, respectively. Moreover, itcan be induced an exact structure through these equivalences and we analyzewhen such quotients are weakly idempotent complete, Krull-Schmidt or abelian.The above discussion is also considered in the particular case of an$(n+2)$-cluster tilting subcategory of $\mathcal{C}$ since in this case we knowthat $\mathcal{X}_{n+1}^{\vee}=\mathcal{C}=\mathcal{X}_{n+1}^{\wedge}.$Finally, by considering the category of conflations of a exact category, weshow that it is possible to get an abelian category from these quotients.

Quotient categories with exact structure from $(n+2)$-rigid subcategories in extriangulated categories