  Let $G$ be a graph in which each edge is assigned one of the colours $1, 2,
\ldots, m$, and let $\Gamma$ be a subgroup of $S_m$. The operation of switching
at a vertex $x$ of $G$ with respect to an element $\pi$ of $\Gamma$ permutes
the colours of the edges incident with $x$ according to $\pi$. We investigate
the complexity of whether there exists a sequence of switches that transforms a
given $m$-edge coloured graph $G$ so that it has a colour-preserving
homomorphism to a fixed $m$-edge coloured graph $H$ and give a dichotomy
theorem in the case that $\Gamma$ acts transitively.
