  A nut graph is a simple graph for which the adjacency matrix has a single
zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry.
If the isolated vertex is excluded as trivial, nut graphs have seven or more
vertices; they are connected, non-bipartite, and have no leaves. It is shown
that a nut graph $G$ always has at least one more edge orbit than it has vertex
orbits: $o_e(G) \geq o_v(G) + 1$, with the obvious corollary that edge
transitive nut graphs do not exist. We give infinite familes of
vertex-transitive nut graphs with two orbits of edges, and infinite families of
nut graphs with two orbits of vertices and three of edges. Several
constructions for nut graphs from smaller starting graphs are known: double
subdivision of a bridge, four-fold subdivision of an edge, a construction for
extrusion of a vertex with preservation of the degree sequence. To these we add
multiplier constructions that yield nut graphs from regular (not necessarily
nut graph) parents. In general, constructions can have different effects on
automorphism group and counts of vertex and edge orbits, but in the case where
the automorphism group is `preserved', they can be used in a predictable way to
control vertex and edge orbit numbers.
