The following question arises naturally in physical contexts when you measure things in a system of certain symmetry $G$ which eventually becomes distorted to lower symmetry $H_a$. There is plenty of information and research on this process but afaics no thorough mathematical description of it, which is parsed smoothly by a mathematician. Typical terms in this context are "Peirls distortion" or "Jahn-Teller effect". In attempting such a formulation, the below formulated question occurs. It takes however some effort to formulate the whole setting as I have experienced, so I must start with a:
Definition (epikernel and kernel):
The epikernel of a group $G$ with respect to an irreducible subspace of $V^{(i)}$ of a representation $V$ of $G$ is $$ \mathrm{Epi}(G,V^{(i)}):=\{\mathrm{Stab_G}(v)| v\in V^{(i)}\},$$ and the kernel $\mathrm{Ke}(G,V^{(i)})$ is the intersection of all elements of this respective epikernel and $\mathrm{Stab_G}(v)$ is the stabilizer group of $v$.
We can define a (non-surjective) map from the set containing $G$ onto the set of all subgroups of $G$
$$ \varphi_{i}: G\mapsto H_{a}\in \mathrm{Epi}(G,V^{(i)}), $$ which is either well defined as it is (if the epikernel contains only one set) or in other cases we can choose one element $H_{a}$. In the latter case there is a bijection from $H_a$ to the maximal subspace $V_a^{(i)}\subset V^{(i)}, $ for which we have $$ \{H_a\} = \{\mathrm{Stab_G}(v)| v\in V_a^{(i)}\},$$
$\varphi_i$ is a map from $G$ to the set of its subgroups corresponding to the irreducible representation with Index $i$. This map is in the context of symmetry-breaking (Jahn-Teller distortion) also called descent in symmetry from $G$ to $H_a$ in direction of $V_a^{(i)}$ (or simply distortion in $V_a^{(i)}$-direction).
The question now is if it is possible to track the action of $\varphi_i$ on the irreducible representations when going from $G$ to $H_a$? Namely:
Is there a generic relation between the irreps of $G$ and $H_a$ in the distortion, which indicates which irrep becomes which or came from which?
and in particular
Can we trace such a relation in the case of a "splitting" of a non-one-dimensional or complex irrep of $G$ into lower-dimensional irreps of $H_a$, in case this takes place.
I know that there will be difficulties, such as for example in
$$\mathrm{Epi}(D_4,E)=\{C_2^{(a)},C_2^{(b)}\} $$
we yield two differently oriented $C_2$ groups I have called these orientations $(a)$ and $(b)$, but here it turns out that in both cases $E$ splits into $A\oplus B$. (While in the standard orientation, that is, the main axis of $D_4$ and $C_2$ oriented in the same direction $B\oplus B $ would combine to $E$ when adding elements to $C_2$ to get $D_4$). So in this example the relation we search for would be $$E[D_4]\to(A,B)[C_2].$$ On the other side physical intuition clearly shows that such a tracing of irreps during distortion must be possible (that argument should hold at least for subgroups of SO(3)).
Any suggestiones?