It is known that $L\models 2^\kappa=\kappa^+$, and that for a set of ordinals $A$ we know that $L[A]\models \exists\lambda\forall\kappa>\lambda(2^\kappa=\kappa^+)$.

In this sense, there is some similarity between $L$ and $L[A]$. Both models have definable well orderings, and both models have a very nice sense of minimality deep within them.

Assuming $0^\sharp$ exists we have the class of Silver indiscernibles from which can define $L$ (using the definable Skolem functions, and the Skolem hull of $I$). Assuming that $A^\sharp$ exists we have a similar class for $L[A]$ as well.

Denote by $I$ the Silver indiscernibles and $I_A$ the corresponding class of $L[A]$.

Is there any intersection between them? Is there some $\alpha$ such that $I\setminus\alpha=I_A\setminus\alpha$?

If the answer to both questions is no in the general case, can we say anything on particular cases known?