We prove a general fixed point theorem for inverse transducers, which implies that the fixed point subgroup of an endomorphism of a finitely generated virtually free group is finitely generated. Furthermore, if the endomorphism is uniformly continuous for the hyperbolic metric, we can prove that the set of regular fixed points in the hyperbolic boundary has finitely many orbits under the action of the finite fixed points. In the automorphism case, it is shown that these regular fixed points are either exponentially stable attractors or exponentially stable repellers. These three theorems were proved for free groups, respectively, by Goldstein and Turner (1986); Cooper (1987); Gabouriau, Jaeger, Levitt and Lustig (1998).