  We consider a family of surfaces of general type $S$ with $K_S$ ample, having
$K^2_S = 24, p_g (S) = 6, q(S)=0$. We prove that for these surfaces the
canonical system is base point free and yields an embedding $\Phi_1 : S
\rightarrow \mathbb{P}^5$.
  This result answers a question posed by G. and M. Kapustka.
  We discuss some related open problems, concerning also the case $p_g(S) = 5$,
where one requires the canonical map to be birational onto its image.
