  We show that for any countable discrete maximally almost periodic group $G$
and any UHF algebra $A$, there exists a strongly outer product type action
$\alpha$ of $G$ on $A$. We also show the existence of countable discrete almost
abelian group actions with a certain Rokhlin property on the universal UHF
algebra. Consequently we get many examples of unital separable simple nuclear
$C^*$-algebras with tracial rank zero and a unique tracial state appearing as
crossed products.
