Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that $\pi(g\cdot m)=\pi(m)$. It is clear that $G$ acts smoothly on each fiber $M_b$ for $b\in B$.
Noe fix a $g\in G$. For each $b\in B$ we consider the fixed point submanifold $M_b^g\subset M_b$.
My question is: when $b$ varies, does the diffeomorphic type of $M_b^g$ unchanged?