  We prove the existence of HK density function for a pair $(R, I)$, where $R$
is a ${\mathbb N}$-graded domain of finite type over a perfect field and
$I\subset R$ is a graded ideal of finite colength. This generalizes our earlier
result where one proves the existence of such a function for a pair $(R, I)$,
where, in addition $R$ is standard graded.
  As one of the consequences we show that if $G$ is a finite group scheme
acting linearly on a polynomial ring $R$ of dimension $d$ then the HK density
function $f_{R^G, {\bf m}_G}$, of the pair $(R^G, {\bf m}_G)$, is a piecewise
polynomial function of degree $d-1$.
  We also compute the HK density functions for $(R^G, {\bf m}_G)$, where
$G\subset SL_2(k)$ is a finite group acting linearly on the ring $k[X, Y]$.
