This is a follow-up on my previous question. So, at the risk of repeating myself, let me give some notation:
Let $k$ be a finite field of characteristic $p\neq 2$ (in fact, one only needs to consider the case $p\in\{3,5\}$), let $\Sigma$ be a finite set of primes containing $\infty$ and $p$, and
$$\rho_{0}:{\rm Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q})\rightarrow {\rm GL}_{2}(k)$$
an absolutely irreducible representation, meaning $\rho_{0}\otimes\overline{k}$ cannot be written as the direct sum of two one-dimensional subrepresentations, where $\mathbb{Q}_{\Sigma}$ is the largest Galois extension of $\mathbb{Q}$ unramified outside the primes in $\Sigma$. Assume that if $\tau\in\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q})$ is complex conjugation, then $\det(\rho_{0}(\tau))=-1$. Then $\rho$ induces a projective representation
$$\tilde{\rho}_{0}:\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q})\longrightarrow\operatorname{PGL}_{2}(k).$$
Assume in this case that the image of the projective representation is $$\operatorname{image}(\tilde{\rho}_{0})\cong\left<s,r\mid s^{2}=r^{2}=(sr)^{2}=1\right>=(\mathbb{Z}/2\mathbb{Z})^{2}\text{.}$$ Assume further that $\rho_{0}|_{L}$ is absolutely irreducible, where $L=\mathbb{Q}(\sqrt{(-1)^{\frac{p-1}{2}}p})$.
The action of $\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q})$ on $k^{2}$ induces an action on $V_{\lambda}=\operatorname{Hom}(k^{2},k^{2})$, namely by conjugation. (What that $\lambda$ stands for is of no significance here.) Since $p\neq 2$, one has a direct sum of $k[\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q})]$-modules
$$V_{\lambda}=W_{\lambda}\oplus k\text{,}$$
where $W_{\lambda}$ denotes the space of $\operatorname{trace}$-$0$ matrices and $k$ is the space of scalar multiplications.
Let $K_{1}$ be the splitting field of $\rho_{0}$ (i.e. $\operatorname{Gal}(\mathbb{Q}_{\Sigma}/K_{1})=\operatorname{ker}(\rho_{0})$), and $$G:=\operatorname{Gal}(K_{1}/\mathbb{Q})=\operatorname{Gal}(\mathbb{Q}_{\Sigma}/\mathbb{Q})/\operatorname{ker}(\rho_{0})=\operatorname{image}(\rho_{0}).$$
Since $\overline{\rho}_{0}$ has dihedral image, $\rho_{0}\otimes\overline{k}=\operatorname{Ind}_{H}^{G}(\chi)$ for some character $\chi$.
As User:Mindlack has generously explained to me in my previous question, $W_{\lambda}$ decomposes as a direct sum of $\operatorname{Gal}(k[\mathbb{Q}_{\Sigma}/\mathbb{Q}]$-modules:
$$W_{\lambda}\otimes\overline{k}=\delta\oplus\operatorname{Ind}_{H}^{G}(\chi/\chi')$$ where $\chi'$ is the quadratic twist of $\chi$ by any element of $G\setminus H$ and $\delta$ is the quadratic character $G\longrightarrow G/H$.
The goal is to show that for any $n\in\mathbb{N}$ and any Galois-stable subspace $X\subseteq W_{\lambda}\otimes\overline{k}$ there exists a $\sigma\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ so that
(i) $\rho_{0}(\sigma)\neq 1$,
(ii) $\sigma$ fixes $\mathbb{Q}(\zeta_{p^{n}})$ and
(iii) $\sigma$ has an eigenvalue $1$ on $X$.
I do not understand Wiles's argument, which runs as follows:
If $\operatorname{image}(\rho_{0})\cong(\mathbb{Z}/2\mathbb{Z})^{2}$, we use the irreducibility of $\rho_{0}|_{L}$. If $\operatorname{Ind}(\chi/\chi')$ is irreducible over $\overline k$ (with $\chi'$ as in my previous question), then $W_{\lambda}\otimes\overline{k}$ is a sum of three distinct quadratic characters, none of which is the quadratic character associated to $L$, and we can repeat the argument by changing the choice of $H$ for the other two characters. If $X=\operatorname{Ind}^{G}_{H}(\chi/\chi')\otimes\overline{k}$ is absolutely irreducible then pick any $\sigma\in G\setminus H$. This satisfies (i) and can be made to satisfy (ii) if $\rho_{0}|_{L}$ is absolutely irreducible.
This is entirely unclear to me. First of all, he seems to make a case distinction where both cases are equal (namely in both cases $\operatorname{Ind}^{G}_{H}(\chi/\chi')$ is absolutely irreducible). My gut feeling tells me it should read reducible in the first case, but even then I don't understand the argument. Also, in the second case, why can one choose $\sigma$ so that it satisfies $(ii)$?
REMARK: I am fully aware that mathoverflow is usually the better place for questions related to the modularity theorem, but this is actually nothing more than straightforward linear algebra with a little Galois action.