Compactness conditions for groups of automorphisms of topological groups

It is proved that if $G$ is a compact, totally disconnected Abelian group and $\operatorname{Aut}G$ is its group of topological automorphisms (with the natural topology), then the following conditions are equivalent: (a) $\operatorname{Aut}G$ is compact; (b) $\operatorname{Aut}G$ is locally compact; (c) $\operatorname{Aut}G$ has small invariant neighborhoods of the identity; (d) $\operatorname{Aut}G$ is an $\overline{FC}$-group; (e) the factor group of $\operatorname{Aut}G$ by its center is compact; (f) the closure of the commutator subgroup of $\operatorname{Aut}G$ is compact; (g) $G\cong\Pi_p(F_p\oplus\Pi_{i=1}^{n_p}Z_p)$, where $F_p$ is a finite $p$-group, $Z_p$ is the additive group of $p$-adic integers, and $n_p<\infty$.