On the Mapping class group of nontrivial $S^2$ fiber bundles

Let $\mathrm{\Sigma }$ be an orientbale closed surface and let ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ be a nonorientable closed surface. In the paper, we show that for any nontrivial orientable $S^2$ fiber bundles $X=\mathrm{\Sigma }⋉S^2$ and $X^{\mathrm{\prime }}={\mathrm{\Sigma }}^{\mathrm{\prime }}⋉S^2$, there are surjective homomorphisms from both $MCG(X)$ and $MCG(X^{\mathrm{\prime }})$ to ${\mathbb{Z}}^{\mathrm{\infty }}$. The proof is an application of generalization of Dax invariants for embedded surfaces in 4-manifolds. The property of $MCG(X)$ and $MCG(X^{\mathrm{\prime }})$ inherits from trivial fiber bundle $\mathrm{\Sigma }\times S^2$.