Let $(R, \mathfrak{m})$ be a Gorenstein local ring of characteistic $p>0$. Let $x\_1,...,x\_d$ be a system of parameters of $R$. Let $I$ be an $\mathfrak{m}$-primary ideal containg $(x\_1,...,x\_d)$. Then there exists an ideal $J$ containng $(x\_1,...,x\_d)$ such that
$(x\_1,...,x\_d):J = I$.


Question. Is it true that for all power of $p$, say $q$, we have
$$(x\_1^q,...,x\_d^q):J^{[q]} = I^{[q]}?$$



Colon property of Gorenstein rings
I am pretty sure that you already know a result of Hochster-Huneke, Corollary 4.3 in "Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem", Journal of the AMS, 1990, for regular local rings. 


I don't think it is true in general. Let $R = \mathbb{Z}/3[X,Y]\_{(X,Y)}/(X^4)$. Let $x,y$ denote the images of $X,Y$ in $R$. Let $m = I = (x,y)$ and $y$ be a system of parameters of $R$. Then $$(y)^{[3]}: m^{[3]} = (y^3): (x^3,y^3) = (y^3) : x^3 = (x,y^3).$$ Since $(x,y^3) \subseteq m \setminus m^2$, the ideal $(x,y^3)$ can not be expressed as the 3rd bracket power of an ideal. 


-------- An example when $R$ is reduced   

I think this works.
One can take $R= \mathbb{Z}/3[X,Y]\_{(X,Y)}/ (X^2-Y^2)$, $I = (x,y)$, and $y$. Then we have $$(y^3) : (x^3,y^3) = (y^3) : x^3 = (y^3) : xy^2.$$ Therefore, $y$ is in the colon ideal. In fact, it is $(x,y)$. By the same reason as above it can not be expressed as the 3rd bracket power of an ideal.


