at.algebraic topology - Action of the mapping class group on middle-dimensional cohomology

Without other assumptions, the answer is an easy no. For instance, if $N^3$ is a homology 3-sphere with infinite fundamental group, then $M^6 = N^3 times S^3$ does not have very many lifts of automorphisms of its middle homology, because there exists a degree one map $S^3 to S^3$ but no map $S^3 to N^3$ with non-zero degree. There are also obstructions from other cup products besides the middle one, and from algebraic operations on cohomology other than cup products.

So the question is much more reasonable if $M$ is simply connected (unless it is 2-dimensional) and has no homology other than the middle homology and at the ends. In this case, Tom Boardman says in the comment that Wall and Freedman showed that the answer is yes for homeomorphisms, although they surely assume that $M$ is simply connected. In higher dimensions, I don't know the answer to this restricted question, but I imagine that it could be yes using surgery theory.