  We resolve three interrelated problems on \emph{reduced Kronecker
coefficients} $\overline{g}(\alpha,\beta,\gamma)$. First, we disprove the
\emph{saturation property} which states that
$\overline{g}(N\alpha,N\beta,N\gamma)>0$ implies
$\overline{g}(\alpha,\beta,\gamma)>0$ for all $N>1$. Second, we esimate the
maximal $\overline{g}(\alpha,\beta,\gamma)$, over all
$|\alpha|+|\beta|+|\gamma| = n$. Finally, we show that computing
$\overline{g}(\lambda,\mu,\nu)$ is strongly $\# P$-hard, i.e. $\#P$-hard when
the input $(\lambda,\mu,\nu)$ is in unary.
