  In 2017 Bar-Natan and the first author showed that solutions to the
Kashiwara--Vergne equations are in bijection with certain knot invariants:
homomorphic expansions of welded foams. Welded foams are a class of knotted
tubes in $\mathbb{R}^4$, which can be finitely presented algebraically as a
circuit algebra, or, equivalently, a wheeled prop. In this paper we describe
the Kashiwara-Vergne groups $\mathsf{KV}$ and $\mathsf{KRV}$ -- the symmetry
groups of Kashiwara-Vergne solutions -- as automorphisms of the completed
circuit algebras of welded foams, and their associated graded circuit algebra
of arrow diagrams, respectively. Finally, we provide a description of the
graded Grothendieck-Teichm\"uller group $\mathsf{GRT}_1$ as automorphisms of
arrow diagrams.
