Determine the isomorphism classes of the representations of $S_4$ induced by $(\text i)$ the one-dimensional representation of the group generated by $(1234)$ in which $(1234)·v = iv,\ $ $i = \sqrt {-1}$; $(\text {ii})$ the one-dimensional representation of the group generated by $(123)$ in which $(123) \cdot v = e^{\frac {2 \pi i} {3}} v.$
By Frobenius reciprocity law I find that the induced representation for $(\text {i})$ and $\text {(ii})$ are respectively $V \oplus (U' \otimes V)$ and $V \oplus (U' \otimes V) \oplus W,$ where $V, U', W$ are respectively the standard representation of $S_4$, alternating representation of $S_4$ and the only irreducible representation of $S_4$ of degree $2.$ Is this the thing the question tells us to do? Can anybody please shed some light on it?
Thanks for your time.
Source $:$ Fulton and Harris Exercise $3.23.$