Induction-isotypical subgroup

Definition with symbols
Let $$H$$ be a subgroup of a group $$G$$ and let $$k$$ be a field. We say that $$H$$ is induction-isotypical in $$G$$ with respect to $$k$$, if, for every irreducible representation $$\sigma$$ of $$H$$, the induction of $$\sigma$$ to $$G$$ is an isotypical representation of $$G$$.

When the field is sufficiently large
In case of a sufficiently large field for the group, the subgroup property of being induction-isotypical is equivalent to the subgroup property of being a conjugacy-closed normal subgroup. This follows as a consequence of the conjugacy class-character duality.

In the general non-modular case
In the general non-modular case, the subgroup property of being induction-isotypical is equivalent to the subgroup property of being a normal subgroup with the property that all Galois-class automorphisms restrict to Galois-class automorphisms (also called a Galois-conjugacy-closed normal subgroup).

In the general (possibly modular) case
In the modular case, it is no longer true that the character determines the representation. Hence, being conjugacy-closed may not suffice for being induction-isotypical. However, every central factor is always induction-isotypical regardless of the field under consideration.