Until someone suggests better terminology, let me call a subgroup H of a finite group G *segregated* if every class function on H can be extended to a class function on G. Equivalently, H should have the property that any two of its elements which happen to be conjugate in G should be conjugate in H.

(This seems to have something to do with the known notions of fusion of conjugacy classes for subgroups, but whenever I've looked in the literature on such things I've not quite found what I was after.)

Here are the few things I know right now:

An obvious example where this happens is the usual inclusion of $S_m \hookrightarrow S_{m+n}$ where we think of $S_m$ as permutations of $\{1,\dots, m+n\}$ that fix each element of $\{m+1,\dots, m+n\}$.

An obvious example where this doesn't happen is the inclusion $A_m \subset S_m$ since one can always find even permutations of the same cycle type which are not conjugate in $A_m$.

If H is contained in Z(G) then it is a segregated subgroup of G.

Malnormal subgroups (i.e. Frobenius complements) are segregated; thus one can have abelian segregated subgroups which aren't central.

Question 1. Suppose G is a finite group with a proper subgroup that is non-abelian. Does it contain a proper subgroup that is non-abelian and segregated?

and

Question 2. Suppose G is a finite group with non-trivial centre. Does it contain a proper subgroup which is non-abelian and segregated?

These questions are motivated by the study of certain Banach algebra norms one can put on the algebra of class functions (with pointwise product), and certain invariants one can associate to these Banach algebras, which do not increase if one passes to quotient algebras. So one would like to take ${\mathcal C}\ell$(G) and restrict to a subgroup H, but this only works well for what I want if the image of the restriction map is all of ${\mathcal C}\ell$(H), i.e. when H is segregated in the sense described above.