Quasi-socle ideals in Gorenstein numerical semigroup rings

Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$ in Gorenstein numerical semigroup rings over fields are explored, where $Q$ is a parameter ideal, and $\mathfrak{m}$ is the maximal ideal in the base local ring, and $q \geq 1$ is an integer. The problems of when $I$ is integral over $Q$ and of when the associated graded ring $\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ of $I$ is Cohen-Macaulay are studied. The problems are rather wild; examples are given.