Group automorphisms with few and with many periodic points

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