Consider the (four) one-dimensional representations of $V_4\simeq S_2\times S_2$ the Klein four group, as listed for example here. It is convenient to consider these as four vectors forming a basis for $\mathbb{Q}^4$ (no need for complex here): $$\left\{\begin{bmatrix}1\\1\\1\\1\end{bmatrix},\begin{bmatrix}1\\1\\-1\\-1\end{bmatrix},\begin{bmatrix}1\\-1\\1\\-1\end{bmatrix},\begin{bmatrix}1\\-1\\-1\\1\end{bmatrix}\right\}$$ Here the action is such that $((1,2),e)$ swaps the first two rows with the bottom two, while $(e,(1,2))$ swaps the 1st/3rd with the 2nd/4th (keeping order otherwise). So $\mathbb{Q}^4$ is a $V_4$-module under this action.
An interesting feature of this is that if one considers $V_4 \triangleleft D_4$ as a (normal) subgroup, one also has a very closely related $D_4$ action on this same set of vectors. It's particularly nice if we recall that $D_4 \simeq S_2\wr S_2$ and so can identify $((e,e);(1,2))$ as an action which leaves the 1st/4th rows alone, but swaps the 2nd and 3rd rows with each other.
Notice that the first and fourth vectors still generate (irreducible) representations, but now the second and third ones do not; instead, somehow they "join" into one irreducible representation (of dimension 2).
Now, this is basically the inverse operation to this very standard question. But I regret to say that we covered induced representations far too hurriedly back in the day, and so I still have a hole in my knowledge here.
Question: Is there any theorem governing when this sort of "merging"/unsplitting happens? Also, is there a word for it? My understanding is that you can't just arbitrarily assume representations of a subgroup magically "work" for the whole group.
Probably wrong thoughts: Maybe this is related to induced representations $\text{Ind}_{V_4}^{D_4}W$, but aren't induced representations twice the dimensionality (in this case, with an index two subgroup), and in that case does that mean both the 2nd and 3rd 1-dimensional reps induce to the same thing? And how would that relate to $\text{Ind}_{V_4}^{D_4}\mathbb{Q}^4$, the induced representation on this whole thing? I do not really understand how to write down induced reps in this concrete way.
Related grumble: Part of it is that representation theory has a slew of words for representations of subgroups becoming representations of the original groups - one paper I read used "induced", "extension", and "inflation" for three separate concepts! Any help in finding a canonical word for this (and how it relates to induced representations) and how to compute with it would be appreciated.