The $2^2$-Bockstein is $\beta_4$ is associated to $$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$

(The $2^n$-Bockstein homomorphism $$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\mathbb{Z}/2)$$ is associated to the short exact sequence $$0\to\mathbb{Z}/2\to\mathbb{Z}/{2^{n+1}}\to\mathbb{Z}/{2^n}\to 0.$$ Note $\beta_2=Sq^1$ is the Steenrod square.)

Question: What are some closed 5-dimensional manifold $M$ satisfy all the criteria below:

1) $M$ is a non-spin manifold.

2) $\beta_{4}$ is nonzero. $$\beta_{4}:H^1(M,\mathbb{Z}/{4})\to H^{2}(M,\mathbb{Z}/2).$$

3) There exists a *non-zero* generator $a \in H^1(M,\mathbb{Z}/2)$, such that its Poincare dual PD$(a)$ is an orientable 4-manifold.

If so, what is this sub-manfiold generator $H^1(M,\mathbb{Z}/2)$ and what is the this orientable 4-manifold PD$(a)$? What is this $M$?

Note that the $\mathbb{RP}^5$ satisfies 1) and 2), but it does not satisfy 3), because the PD$(a)$ for $\mathbb{RP}^5$ is a non-orientable 4-manifold $\mathbb{RP}^4$.