  Given a triangulated category $D$ with an action of a fusion category $C$, we
study the moduli space $Stab_{C}(D)$ of fusion-equivariant Bridgeland stability
conditions on $D$. The main theorem is that the fusion-equivariant stability
conditions form a closed, complex submanifold of the moduli space of stability
conditions on $D$. As an application of this framework, we generalise a result
of Macr\`{i}--Mehrotra--Stellari by establishing a homeomorphism between the
space of $G$-invariant stability conditions on $D$ and the space of
$rep(G)$-equivariant stability conditions on the equivariant category $D^G$. We
also describe applications to the study of stability conditions associated to
McKay quivers and to geometric stability conditions on free quotients of smooth
projective varieties.
