Given a finite group action on a manifold, we discuss the following question: if two equivariant diffeomorphisms are isotopic, must they be equivariantly isotopic? In the case of closed hyperbolic surfaces, a remarkable theorem of Birman and Hilden says that the answer is “yes”: isotopy implies equivariant isotopy. By contrast, we show that in dimensions three and higher, there are many diffeomorphisms which are isotopic but not equivariantly isotopic. We will explain the new obstructions that arise in higher dimensions, as well as some applications and further questions that don’t arise in the world of surfaces.