Automorphism groups of Boolean powers with ample generics

Let $A$ be a finite non-abelian simple Mal'cev algebra, such as for example a finite simple non-abelian group or a finite simple non-zero ring. We show that the automorphism group of a filtered Boolean power of $A$ by the countable atomless Boolean algebra $A$ has ample generics. This uses the decomposition of that automorphism group as a semidirect product of a certain closure of a Boolean power of the automorphism group of $A$ by $B$ and the stabiliser of finitely many points in the homeomorphism group Homeo$2^\omega$ of the Cantor space $2^\omega$ by the authors. As an intermediate step, we show that pointwise stabilisers in Homeo$2^\omega$ have ample generics, which extends the result of Kwiatkowska that Homeo$2^\omega$ has ample generics.