If $\,D(K)\,$ is an irreducible of $\,K\vartriangleleft G\,$, then $\,D_g(K)\equiv D(g^{-1}Kg)\,$ are irreducibles of $\,K\,$ for $\,\forall~g\in G\,$. The elements $\,h\,$ generating representations equivalent to $\,D\,$, constitute an isotropy subgroup: $$ D_h(K)~\simeq~D(K)\quad\Longleftrightarrow\quad h\in H~<~G $$
Likewise, the elements $\,h^{\,\prime}\in G\,$, which satisfy $$ \left(D_g\right)_{h'}(K)~\simeq~D_g(K)~~, $$ constitute the isotropy group $\,H_g\,$ of $\,D_g(K)\,$. The groups $\,H_g\,$ and $\,H\,$ are conjugate: $$ H_g~=~g~H~g^{-1}~~. $$
So the set of $\,D_g\,$ splits into disjoint classes (orbits) of mutually equivalent irreducibles.
Lemma 1
$\phantom{qqi}$ If $\,H\,$ is the isotropy group of $\,D(K)\,$, then, for $~\forall\,w,\,w^{\,\prime}\,\in\, G\,$, $$ D_{w^{~\prime}}(K)~\simeq~D_w(K)\quad\iff\quad w^{~\prime}~=~w~h~~,~~h\in H~~. $$ This can be easily proven from the definition of $\,H\,$.
Lemma 2
$\phantom{qqi}$ For $~\forall~w,~w^{~\prime} \in G~$ and $~\forall h \in H\,$, $$ w^{~\prime}~=~w~h\quad\Longrightarrow\quad H~w^{~\prime}~H~=~H~w~H~~. $$ $\phantom{qqi}$ In the special case of $H \lhd G $, $$ w^{~\prime}~=~w~h\quad\iff\quad H~w^{~\prime}~H~=~H~w~H~~. $$ The former statement is self-evident. To prove the latter one, i.e. to show that for $H\lhd G$ the implication becomes equivalence, we notice that in this situation, for $\forall h_1,~h_2\in H$ and $\forall w \in G$, a product $ h_1 w h_2 $ can be cast as $$ h_1~ w~h_2~=~w~(w^{-1}~h_1~ w)~h_2~=~w~h~~,~~~ h\in H~~. $$ Hence, $$ H~w^{~\prime}~H~=~H~w~H \quad\iff\quad w^{~\prime}~=~h_1~w~h_2 \quad\iff\quad w^{~\prime}~=~w~h~~. $$ QED
Combined, the former and latter lemmas amount to the following
Theorem $$ D_{w^{\,\prime}}(K)~\simeq~D_w(K)\quad\iff\quad w^{~\prime}~=~w~h\quad\Longrightarrow\quad H\,w^{~\prime}~H~=~H~w~H~~. $$ $\phantom{qqi}$ In the special case of $H \lhd G $, this works in both directions: $$ D_{w^{~\prime}}(K)~\simeq~D_w(K)\quad\iff\quad w^{~\prime}~=~w~h\quad\iff\quad H\,w^{~\prime}~H~=~H~w~H~~.\quad $$
Thus, if for some $\,K\lhd G\,$ we start out with a representation $\,D(K)\,$ and its isotropy group $\,H\,$, and build the space $\,H\backslash G/H\,$, then distinct double cosets $~H w H~$ and $~H w^{~\prime} H~$ parameterise inequivalent representations $D_w(K)$ and $D_{w^{~\prime}}(K)$ $-$ and, thereby, different orbits.
While in general it is not prohibited for one double coset to correspond to several inequivalent irreducibles (i.e. to several orbits), in the case of $H \lhd G $ the mapping becomes bijective, so one coset corresponds to one orbit.
QUESTION:
Can you please offer me a simple example of a situation where $w$ ranges within one coset $HwH$ and generates several inequivalent $D_w(K)~$?