Abstract Semiprojectivity is a lifting property for C*-algebras that has proved useful in the classification program. Phillips has recently introduced an equivariant version of semiprojectivity, and given many examples of semiprojective actions by compact groups. We will discuss the relations between semiprojectivity with and without a group action. For instance, we will see that if a compact group acts semiprojectively on a C*-algebra A, than A must be (ordinarily) semiprojective, and that there is a semiprojective action of the group of integers on a non-semiprojective C*-algebra. This is joint work with N. Christopher Phillips and Hannes Thiel.