How can I compute the characters of the induced representation $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$?
Here, $S_n$ is the symmetric group over $n$ symbols and $D_n$ is the dihedral group of order $2 n$.
I can compute the character table of the symmetric group $S_n$ using the scheme mentioned in this blog article which uses the Murnaghan-Nakayama rule.
Is there a similar approach for computing the characters of $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$?
I understand that I can compute the transversal of $D_n$ in $S_n$ and use it to compute the matrix representation of each element in $S_n$ as shown in the section 1.12 of The Symmetric Group : Representations, Combinatorial Algorithms, and Symmetric Functions by Sagan. Then I can compute the characters by taking the traces. But this technique is tedious.
Is there any shorter procedure where I can use the Murnaghan-Nakayama rule to compute the characters of $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$?
Motivation
I am trying to detect the hidden subgroup $D_n$ of order $2 n$ within $S_n$ using Normal Subgroup Reconstruction and Quantum Computation Using Group Representations as reference. This boils down to computing the Fourier transform of the following indicator function of a left coset of $D_n$ in $S_n$ which is assumed.
$$ f (g) = \begin{cases} \frac{1}{\sqrt{|D_n|}} & \quad \text{ if } g \in c D_n\\ 0 & \quad \text{ otherwise}\\ \end{cases} $$ for some $c \in S_n$.
If the quantum Fourier transform is $\hat{f}$, we seek to measure the labels of irreps, $\rho$. According to the Lemma 1 in the paper, the probability of measuring $\rho$ is $\frac{|D_n|}{|S_n|} d_\rho \langle \chi_\rho, \chi_{\mathbf{1}_H}\rangle_{D_n}$.
Using a special case of Frobenius reciprocity as shown in the Lemma 2 in the paper, we can say that, $$\langle {\chi \uparrow^{S_n}_{D_n}}_{\mathbf{ 1}_{D_n}} , \chi_\rho \rangle_{S_n} = \langle \chi_{\mathbf{ 1}_{D_n}}, \chi_\rho \downarrow^{S_n}_{D_n} \rangle_{D_n}$$
That's why I need to compute $\langle {\chi \uparrow^{S_n}_{D_n}}_{\mathbf{ 1}_{D_n}} , \chi_\rho \rangle_{S_n}$. To do that I need to determine the character table of $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$.