For a cyclic covering map (Σ,K) → (Σ',K') between two pairs of a 3-manifold and a knot each, we describe the fundamental group π_1(Σ∖ K) in terms of π_1(Σ' ∖ K'). As a consequence, we give an alternative proof for the fact that certain knots in S^3 cannot be represented as the preimage of any knot in a lens space, which is related to free periods of knots. In our proofs, the subgroup of a group G generated by the commutators and the pth power of each element of G plays a key role.