Let $H$ be a Hopf $\pi$-coalgebra and let $A$ be a right group $H$-comodule algebra with a total integral $\phi$. In this article we will find some sufficient conditions under which an epimorphism of $(H, A)$-Hopf $\pi$-comodule splits if it splits $A$-linearly. As an application, we obtain a characterization for an $(H, A)$-Hopf $\pi$-comodule to be projective as an $A$-module, generalizing the one of the Maschke type theorem found in [D2].
"A Maschke type theorem for Hopf $\pi$-comodules." Tsukuba J. Math. 28 (2) 377 - 388, December 2004. https://doi.org/10.21099/tkbjm/1496164806