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=SymV be the symmetric algebra with the standard grading. Let M be a 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∖F is free. Generalizing the result of P. Symonds and the first author, we describe the Frobenius limit FL(LG) of the SG-module LG. In particular, we determine the generalized F-signature s(M,SG) for each indecomposable gradable reflexive SG-module M. In particular, we prove the fact that the F-signature s(SG)=s(SG,SG) equals 1/dimk[G] if G is linearly reductive (already proved by Watanabe–Yoshida, Carvajal-Rojas–Schwede–Tucker, and Carvajal-Rojas) and 0 otherwise (some important cases have already been proved by Broer, Yasuda, Liedtke–Martin–Matsumoto).