Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles

Let $F^{\mathrm{\prime }},F$ be any two closed orientable surfaces of genus $g^{\mathrm{\prime }}>g\ge 1$, and $f:F\to F$ be any pseudo-Anosov map. Then we can “extend” $f$ to be a pseudo-Anosov map $f^{\mathrm{\prime }}:F^{\mathrm{\prime }}\to F^{\mathrm{\prime }}$ so that there is a fiber preserving degree one map $M(F^{\mathrm{\prime }},f^{\mathrm{\prime }})\to M(F,f)$ between the hyperbolic surface bundles. Moreover the extension $f^{\mathrm{\prime }}$ can be chosen so that the surface bundles $M(F^{\mathrm{\prime }},f^{\mathrm{\prime }})$ and $M(F,f)$ have the same first Betti numbers.