Suppose I am computing the (co)homology of a nice space M using a local coefficient system G, i.e. $H_{*}(M, G)$. If M is homotopy equivalent to N, then M and N should have isomorphic (co)homology. Suppose $f: M \to N$ is the relevant map between nice spaces. My question is this: how can I use knowledge about the homology of N to compute homology of M with the local coefficient system G? Basically, I do not know what coefficient system f would induce on N.
To make the situation more specific, I am actually working with a fibration $F \to E \to M$ and am computing homology of M with coefficients the fiber F, so $H_{*}(M, H_{0}(F, \mathbb{C}))$. However, it is easy enough for me to show M is homotopy equivalent to a (finite) wedge product of a nice space. Say there are n of them. In this case, if my coefficient system on the nice space is P, nth homology of my nice wedge is n copies of P and everything works smoothly. However, I really have no idea to deduce what P would look like from data about M and F.
Any help would be greatly appreciated.
(A bonus, but much less important question, is this: if $M_{0}$ and $M_{1}$ are homotopy equivalent and I have the fibration $F_{0} \to E_{0} \to M_{0}$ do the relevant homotopy maps $f: M_{0} \to M_{1}, g: M_{1} \to M_{0}$ induce a compatible map of fibrations? By compatible I mean something like constructing a fibration of $M_{1}$ with some commutativity involving f and/or g.