Let $G$ be a graph in which each edge is assigned one of the colours $1, 2, \ldots, m$, and let $Γ$ be a subgroup of $S_m$. The operation of switching at a vertex $x$ of $G$ with respect to an element $π$ of $Γ$ permutes the colours of the edges incident with $x$ according to $π$. 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 $Γ$ acts transitively.