incompressible one - sided surfaces

Question

Suppose we are given an orientable 3 - manifold M and an embedded closed and one - sided surface S with normal bundle N. It is well known that $\delta N$ is an orientable subsurface covering S. Suppose further that $\delta N$ is incompressible. Does this imply that S is incompressible and, moreover, that $i_*: \pi_1 (S) \to \pi_1 (M) $ is injective ?