The spectra of cardinalities of branches of Kurepa trees, part 1

We study the possible values of the Kurepa spectra, i.e. how the set of cardinalities of branches of Kurepa trees may look like. It turns out that assuming GCH below the second uncountable cardinal (and possibly the existence of an inaccessible) we can force every set of cardinals (satisfying some obvious necessary conditions) to be the Kurepa spectrum.