We give a classification of finite group actions on a $K3$ surface giving rise to $K3$ quotients, from the point of view of their fixed points. It is shown that except two cases, each such group gives rise to a unique type of fixed point set.

A generalised Weber function is given by ${}_N\left(z\right)=\eta (z/N)/\eta \left(z\right)$, where η(z) is the Dedekind function and N is any integer; the original function corresponds to N=2. We classify the cases where some power ${}_N^e$ evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating ${}_N\left(z\right)$ and j(z). Our ultimate goal is the use of these invariants in constructing...