Group automorphisms with few and with many periodic points

For any $C\in [0,\mathrm{\infty }]$ a compact group automorphism $T:X\to X$ is constructed with the property that $$\frac{1}{n}\log \mathrm{\mid }\{x\in X\mid T^n(x)=x\}\mathrm{\mid }⟶C\mathrm{.}$$ This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms exist with any given topological entropy.