  We further the techniques developed by Etingof, Nikshych, and Ostrik in order
to classify the $\mathcal{C}$-based equivalences between two $G$-graded
extensions of $\mathcal{C}$. The main result of this paper (which follows from
this classification) shows that there is an action of the group
$\operatorname{Aut}(G)\times \operatorname{Aut}_\otimes(\mathcal{C})$ on the
set of all $G$-graded extensions of $\mathcal{C}$, and further, any two
extensions in the same orbit of this action are monoidally equivalent. As a
warm up for the proof of our classification result we reprove the
classification of graded extensions of a fusion category, making extensive use
of graphical calculus. Aside from our main result, we provide several other
practical applications of our classification of $\mathcal{C}$-based
equivalences.
