Let $Z * Z/2Z = \langle a, b | b^2=1\rangle$ be represented by $X = S^1\vee RP^2$ i.e. the wedge of the unit circle and the real projective plane.
Let $H$ be the smallest normal subgroup containing $b$.
Question:
How can we construct a covering space $\tilde X$ corresponding to $H$ by sketching a good picture for $\tilde X$ that covers X.
Any help is really appreciated.
Thanks