Suppose we have a fiber sequence of connected spaces $A\rightarrow B\rightarrow C$ and suppose we know the homology of A and B, is there a homological spectral sequence converging to the homology of $C$.

$\textbf{Edit}:$ I think a closely related question is the following. Suppose we have a map of fiber sequence $[A\rightarrow B\rightarrow C]\rightarrow [A^{'}\rightarrow B^{'}\rightarrow C^{'}]$ where all involved spaces are connected and the maps $A\rightarrow A^{'}$, $B\rightarrow B^{'}$ induce homology isomorphism, could we conclude that the map $C\rightarrow C^{'}$ induces a homology isomorphism.