  For a sequence of the naturally graded quasi-filiform Leibniz algebra of
second type $\mathcal{L}^2$ introduced by Camacho, G\'{o}mez, Gonz\'{a}lez and
Omirov, all possible right and left solvable indecomposable extensions over the
field $\Bbb R$ are constructed so that the algebra serves as the nilradical of
the corresponding solvable Leibniz algebras we find in the paper.
