We first find simple characterizations of 1/NZ-invariance of arbitrary principal shift-invariant space V(phi). Then we find several equivalent conditions for V(phi) to admit periodic oversampling for a class of continuous frame generators phi. In particular, when phi is band-limited and (phi) over cap is piecewise continuous, we find very simple and general sufficient conditions for V(phi) to admit periodic oversampling, which involve the extra invariance of V(phi), together with an illustrating example.