  If $G$ is a group, a virtual retract of $G$ is a subgroup which is a retract
of a finite index subgroup. Most of the paper focuses on two group properties:
property (LR), that all finitely generated subgroups are virtual retracts, and
property (VRC), that all cyclic subgroups are virtual retracts. We study the
permanence of these properties under commensurability, amalgams over retracts,
graph products and wreath products. In particular, we show that (VRC) is stable
under passing to finite index overgroups, while (LR) is not.
  The question whether all finitely generated virtually free groups satisfy
(LR) motivates the remaining part of the paper, studying virtual free factors
of such groups. We give a simple criterion characterizing when a finitely
generated subgroup of a virtually free group is a free factor of a finite index
subgroup. We apply this criterion to settle a conjecture of Brunner and Burns.
