Can the cardinality of the set of all intervening cardinals between sets and their power sets be always singular?

Question

Is the following known to be consistent relative to some large cardinal assumption?

$\forall \kappa [\kappa >2 \to \kappa < \kappa^* < 2^\kappa \wedge singular(\kappa^*)]$

where $\kappa$ is a cardinal and $``<"$ is strict cardinal smaller than, and $\kappa^* = |\{\lambda|\kappa < \lambda < 2^\kappa\}|$