Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $.

What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true that there are only finitely many such complex surfaces (i.e. with a given $\pi_1$ and $\kappa(S) = 1$) up to diffeomorphism?

Clearly, such a statement doesn't hold when such $S$ is a simply connected complex surface with $\kappa(S) = 1$ since elliptic surfaces $E(n; p, q)$ with two logarithmic transformations provide such infinite family of diffemorphism types, where $(p, q) = 1$. Probably, a similar result holds when $\pi_1(S) = \mathbb{Z}_{n}$, but I am not certain when $b_{1}(S) \geq 2$. Any references would be greatly appreciated.

Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $.

What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true that there are only finitely many such complex surfaces (i.e. with a given $\pi_1$ and $\kappa(S) = 1$) up to diffeomorphism?

Clearly, such a statement doesn't hold when such $S$ is a simply connected complex surface with $\kappa(S) = 1$ since elliptic surfaces $E(n; p, q)$ with two logarithmic transformations provide such infinite family of diffemorphism types, where $(p, q) = 1$. Probably, a similar infiniteness result holds when $\pi_1(S) = \mathbb{Z}_{k}$, but I am not sure when $b_{1}(S) \geq 2$. Any references would be greatly appreciated.