A dichotomy theorem for $\mathrm{\Gamma }$-switchable $H$-colouring on $m$-edge coloured graphs

Let $G$ be a graph in which each edge is assigned one of the colours $1,2,\dots ,m$, and let $\mathrm{\Gamma }$ be a subgroup of $S_m$. The operation of switching at a vertex $x$ of $G$ with respect to an element $\pi$ of $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ acts transitively.