We define and develop preliminary theoretical results for the $Γ$-switch Ramsey number, a variation on the classical $m$-colour Ramsey number for which we allow permuting the colours incident with a vertex using elements of a group $Γ\leq S_m$. We find bounds for the $Γ$-switch Ramsey number for groups with various properties as a function of the classical parameter. We prove $R_{C_3}(4,4,4) = R(3,3,3) + 1$, $R_{C_4}(3,4,3,4) = R(2,3,2,3) + 1$ and $43 \leq R_{C_4}(4,4,4,4) \leq R(3,3,3,3) + 1$.