  We provide a systematic description of the automorphism groups of specially
cocompact CAT(0) cube complexes. We show that these groups are topologically
finitely generated, present a method to explicitly obtain generating sets, and
prove a dichotomy on their size. Furthermore, we show that, under some extra
assumptions, the normal subgroup known as Aut^+ is simple, non-discrete, and
tdlc. In particular, we obtain a new class of simple, non-discrete, tdlc groups
that are accessible to further study. Finally, we study the relative size of
Aut^+ in the automorphism group, providing a sufficient condition for its
closure to be finite index and presenting a common example where it is not even
cocompact.
