When are the rings $I:I$ Gorenstein?

Let $I ~(\ne A)$ be an ideal of a $d$-dimensional Noetherian local ring $A$ with $\operatorname{ht}_AI \ge 2$, containing a non-zerodivisor. The problem of when the ring $I:I=\operatorname{End}_AI$ is Gorenstein is studied, in connection with the problem of the Gorensteinness in Rees algebras $\mathcal{R}_A(Q^d)$ for certain parameter ideals $Q$ of $A$, that was closely explored by the preceding paper of the second and third authors. Examples are given.