Let $X$ be a smooth variety and let $Y\subset X$ be a smooth subvariety. Let $f:Y'\to Y$ be a (say, finite surjective) morphism. When $f$ is the identity, the cohomology group $H^0(Y',f^*N_Y)$ measures the first order deformations of $Y$ in $X$. My question is:
Does the group $H^0(Y',f^*N_Y)$ have a deformation-theoretic interpretation for other $f$?
Of course you could say that this group fits into an exact sequence involving $H^0(Y,f^*T_X|_Y)$ which measures deformations of the morphism $f$, but my question is really whether he above group measures something by its own.
I am mostly interested in the cases where $f$ is the Frobenius or the normalization map.