Let p be an odd prime and r be relatively prime to p. Let G
be a finite p–group. Suppose an oriented 3–manifold \tilde
M has a free G–action with orbit space M. We consider
certain Witten–Reshetikhin–Turaev SU(2) invariants
wr(M)
in Z[1/2r,e2πi/8r]. We will show that
wr(~M)≡κ3
def(~M→M)(wr(M))|G| (mod p).
Here κ=e2πi(r-2)/8r,
def denotes the signature defect, and |G| is the number
of elements in G. We also give a version of this result if M and
~M contain framed links or colored fat graphs. We give similar
formulas for non-free actions which hold for a specified finite set of
values for r.
