Given two Hopf algebras $A,B$ over a field $k$, and a Hopf algebra map $\pi:A \to B$, are there any general tricks for finding a complement to the kernel of $\pi$. That is, how can one find a subspace $C$ such that $C \oplus $ker$(\pi) = A$?

P.S. To confirm, by *complement* I mean I simply mean vector space complement.