Let $f:X \rightarrow Y$ be an (unramified) holomorpic covering map between two (maybe non compact) complex manifolds.

**Q:** Does every infinitesimal deformation of Y lift *faithfully* to an infinitesimal deformation of X, (i.e. is there a canonical injective map $l:H^1(Y, \Theta_Y) \rightarrow H^1(X,\Theta_X)$?

If not, do you know a counterexample?

Thanks!