 Solution

Suppose a ground cardinal $\lambda \ge \kappa$ became smaller in the extension. There would be a surjection $f:\alpha \to \lambda$ for some ordinal $\alpha <\lambda$. A finite domain cannot map onto an infinite ordinal, so use part (a) to cover $f$ by a ground function $y$ of small value sets. Its range is contained in $U={\bigcup }_{\zeta <\alpha }y(\zeta )$.

If $\lambda >\kappa$, the ground cardinality of $U$ is at most $|\alpha {|}^M\cdot \kappa <\lambda$. If $\lambda =\kappa$, regularity bounds a union of fewer than $\kappa$ sets of size less than $\kappa$ by a cardinal less than $\kappa$. In either case $U$ is a proper ground subset of $\lambda$, and the same missing ordinal remains missing in the extension, contradicting surjectivity.

Hence all ground cardinalities at least $\kappa$ are preserved. Ground bijections also remain bijections, so old noncardinals cannot turn into cardinals. The assertion does not prevent collapse of smaller cardinals, and consequently does not by itself preserve the aleph index of $\kappa$.