  Let $G$ be a graph whose edges are each assigned one of the $m$-colours $1,
2, \ldots, m$, and let $\Gamma$ be a subgroup of $S_m$. The operation of
switching at a vertex $x$ with respect $\pi \in \Gamma$ permutes the colours of
the edges incident with $x$ according to $\pi$. There is a well-developed
theory of switching when $\Gamma$ is Abelian. Much less is known for
non-Abelian groups. In this paper we consider switching with respect to
non-Abelian groups including symmetric, alternating and dihedral groups. We
first consider the question of whether there is a sequence of switches using
elements of $\Gamma$ that transforms an $m$-edge-coloured graph $G$ to an
$m$-edge coloured graph $H$. Necessary and sufficient conditions for the
existence of such a sequence are given for each of the groups being considered.
We then consider the question of whether an $m$-edge coloured graph can be
switched using elements of $\Gamma$ so that the transformed $m$-edge coloured
graph has a vertex $k$-colouring, or a homomorphism to a fixed $m$-edge
coloured graph $H$. For the groups just mentioned we establish dichotomy
theorems for the complexity of these decision problems. These are the first
dichotomy theorems to be established for colouring or homomorphism problems and
switching with respect to any group other than $S_2$.
