Canonical surfaces of higher degree

We consider a family of surfaces of general type $S$ with $K_S$ ample, having $K_S^2=24,p_g(S)=6,q(S)=0$. 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.