Universal property of framed $G$-disc algebras

Given a compact Lie group $G$ and its finite subgroup $H$ we prove that the $\infty$-category of $G/H$-framed $G$-disc algebras taking values in a $G$-symmetric monoidal category $\underline{\mathcal{C}}^{\otimes}$ is equivalent to the $\infty$-category of $V$-framed $H$-disc algebras (where $V$ is an $H$-representation) which take values in $\underline{\mathcal{C}}^{\otimes}_H$, the underlying $H$-symmetric monoidal subcategory of $\underline{\mathcal{C}}^{\otimes}$. We will use this construction to refine the $C_2$-action on the real topological Hochschild homology to an $O(2)$-action.