Can GAP find isomorphisms to permutation groups which preserve / have additional properties?

Question

Let $G$ and $H$ be two finite non-isomorphic groups in GAP.

The GAP command

IsomorphismPermGroup

can compute permutation groups isomorphic to $G$ and other permutation groups isomorphic to $H$.

Suppose that $G$ and $H$ have isomorphic defect groups $D_G \cong D_H$.

Question:

Is there a way to compute isomorphic permutations groups $\hat{H}$ of $H$ and $\hat{G}$ of $G$ with GAP such that $D_\hat{G}=D_\hat{H}$ are really the same permutation group with the same elements, and not only isomorphic?

Thank you very much for the help.