On Sushchansky $p$-groups

We study Sushchansky $p$-groups introduced in [Sus79]. We recall the original definition and translate it into the language of automata groups. The original actions of Sushchansky groups on $p$-ary tree are not level-transitive and we describe their orbit trees. This allows us to simplify the definition and prove that these groups admit faithful level-transitive actions on the same tree. Certain branch structures in their self-similar closures are established. We provide the connection with, so-called, $\mathsf G$ groups [BGŠ03] that shows that all Sushchansky groups have intermediate growth and allows to obtain an upper bound on their period growth functions.