Before I write my question, I want to write some thoughts.
Let $M$ be a connected topological manifold such that $\pi_1(M)=\Bbb Z/3\Bbb Z$. Then, considering its orientation $2$-fold cover, which is connected, I can say $M$ is orientable. Now, an example of such a closed $3$-manifold is $L(3,1)$.
Now, this type of argument can not be given if I consider $\pi_1(M)=\Bbb Z/4\Bbb Z$ to conclude $M$ is orientable. But Euler characteristic of an odd-dimensional closed manifold is always zero, so we cannot say $\Bbb Z/4\Bbb Z$ is the fundamental group of any closed connected non-orientable $3$-manifold, as $H_1(M,\Bbb Z)$ is infinite when $M$ is closed non-orientable connected $3$-manifold.
Again this logic can not be given for $4$-dimensional closed connected manifold. So, I am wondering if the following fact. I assume closed means compact without boundary.
Does there exist closed connected $4$-manifolds both orientable and non-orientable type having fundamental group $\Bbb Z/4\Bbb Z$?
Any help will be appreciated.