  We compute the dimensions of $\operatorname{Ext}_G^n(V, W)$ for all
irreducible $V$, $W$ lying in $r$-blocks of cyclic defect in the simple groups
$\operatorname{Sz}(q)$, $\operatorname{PSU}_3(q)$ and
$\operatorname{{}^2G}_2(q)$ in cross characteristic, obtaining in particular
the dimensions of all cohomology groups for such modules. Along the way, we
also obtain an analogous result for any $r$-block of cyclic defect whose Brauer
tree is either a star or line (open polygon).
