Which pairs of cardinals can be Hartogs and Lindenbaum numbers of a set?

Given any $\lambda \leq \kappa $, we construct a symmetric extension in which there is a set $X$ such that $\aleph (X)=\lambda $ and $\aleph ^*(X)=\kappa $. Consequently, we show that $\mathsf{ZF} {}+{}$“for all pairs of infinite cardinals $\lambda \leq \kappa $ there is a set $X$ such that $\aleph (X)=\lambda \leq \kappa =\aleph ^*(X)$” is consistent.