  We show how to find a complete set of necessary and sufficient conditions
that solve the fixed-parameter local congruence problem of immersions in
$G$-spaces, whether homogeneous or not, provided that a certain $k^{\rm th}$
order jet bundle over the $G$-space admits a $G$-invariant local coframe field
of constant structure. As a corollary, we note that the differential order of a
minimal complete set of congruence invariants is bounded by $k+1$. We
demonstrate the method by rediscovering the speed and curvature invariants of
Euclidean planar curves, the Schwarzian derivative of holomorphic immersions in
the complex projective line, and equivalents of the first and second
fundamental forms of surfaces in ${\mathbb R}^3$ subject to rotations
