Suppose a finite group acts on a closed manifold M. Given two equivariant homeomorphisms of M, if we know that they are isotopic, can we conclude that they are equivariantly isotopic? An important theorem of Birman-Hilden and MacLachlan-Harvey says the answer is “yes” if M is a hyperbolic surface; Margalit-Winarski asked whether the same is true when M is a 3-manifold. We answer Margalit-Winarski’s question for a wide class of group actions on 3-manifolds; this includes a 3-manifold analog of the hyperelliptic involution, which we can understand particularly well via a connection with geometric group theory.