  Let $G$ be an affine algebraic group over an algebraically closed field $k$
of characteristic zero. In this paper, we consider finite $G$-equivariant
morphisms $F:X\to Y$ of irreducible affine $G$-varieties. First we determine
under which conditions on $Y$ the induced map $F^G:X//G\to Y//G$ of quotient
varieties is also finite. This result is reformulated in terms of kernels of
derivations on $k$-algebras $A\subset B$ such that $B$ is integral over $A$.
Second we construct explicitly two examples of finite $G$-equivariant maps $F$.
In the first one, $F^G$ is quasifinite but not finite. In the second one, $F^G$
is not even quasifinite.
