Extendability of foliations

Given a foliation F on X and an embedding X\subseteq Y, is there a foliation on Y extending F? Using formal methods, we show that this question has an affirmative answer whenever the embedding is sufficiently positive with respect to (X,F) and the singularities of F belong to a certain class. These tools also apply in the case where Y is the total space of a deformation of X. Regarding the uniqueness of the extension, we prove a foliated version of a statement by Fujita and Grauert ensuring the existence of tubular neighborhoods. We also give sufficient conditions for a foliation to have only trivial unfoldings, generalizing a result due to Gómez-Mont.