  Let $k$ be a perfect field of prime characteristic $p$, $G$ a finite group
scheme over $k$, and $V$ a finite-dimensional $G$-module. Let
$S=\mathop{\mathrm{Sym}}V$ be the symmetric algebra with the standard grading.
Let $M$ be a $\Bbb Q$-graded $S$-finite $S$-free $(G,S)$-module, and $L$ be its
$S$-reflexive graded $(G,S)$-submodule.
  Assume that the action of $G$ on $V$ is small in the sense that there exists
some $G$-stable Zariski closed subset $F$ of $V$ of codimension two or more
such that the action of $G$ on $V\setminus F$ is free. Generalizing the result
of P. Symonds and the first author, we describe the Frobenius limit
$\mathop{\mathrm{FL}}(L^G)$ of the $S^G$-module $L^G$. In particular, we
determine the generalized $F$-signature $s(M,S^G)$ for each indecomposable
gradable reflexive $S^G$-module $M$. In particular, we prove the fact that the
$F$-signature $s(S^G)=s(S^G,S^G)$ equals $1/\dim k[G]$ if $G$ is linearly
reductive (already proved by Watanabe--Yoshida,
Carvajal-Rojas--Schwede--Tucker, and Carvajal-Rojas) and $0$ otherwise (some
important cases has already been proved by Broer, Yasuda,
Liedtke--Martin--Matsumoto).
