  For any $n\geq 7$, $k\geq 3$, we give pairs of compact flat $n$-manifolds $M,
M'$ with holonomy groups $\mathbb Z_2^k$, that are strongly isospectral, hence
isospectral on $p$-forms for all values of $p$, having nonisomorphic cohomology
rings. Moreover, if $n$ is even, $M$ is K\"ahler while $M'$ is not.
Furthermore, with the help of a computer program we show the existence of large
Sunada isospectral families; for instance, for $n=24$ and $k=3$ there is a
family of eight compact flat manifolds (four of them K\"ahler) having very
different cohomology rings. In particular, the cardinalities of the sets of
primitive forms are different for all manifolds.
