  We prove that the automorphism group of a compact 6-manifold $M$ endowed with
a symplectic half-flat SU(3)-structure has abelian Lie algebra with dimension
bounded by min$\{5,b_1(M)\}$. Moreover, we study the properties of the
automorphism group action and we discuss relevant examples. In particular, we
provide new complete examples on $T\mathbb{S}^3$ which are invariant under a
cohomogeneity one action of SO(4).
