= Solution
For any <cyclic subgroup> $H=\langle s\rangle$ of order $n\in\{1,2,4\}$, define
$$
\boxed{\rho_1(s^j)=e^{2\pi ij/n}\in GL(1,\mathbb C),\qquad
\rho_2(s^j)=\begin{pmatrix}\cos(2\pi j/n)&-\sin(2\pi j/n)\\\sin(2\pi j/n)&\cos(2\pi j/n)\end{pmatrix}\in GL(2,\mathbb R).}
$$
Both preserve products because scalar phases and planar rotations add their angles. Each sends the generator to an element of exact order $n$, so both have trivial <kernel> and are <faithful representations>.
For $\langle U\rangle\cong C_4$, take $s=U$: the generator images are $i$ and $R(\pi/2)$. For each order-two subgroup take its nonidentity element $s$; the images are $-1$ and $-I_2$. For the <trivial subgroup> the unique images are $1$ and $I_2$, which are faithful on a one-element group.
The one-dimensional representation of $C_4$ here must use complex scalars. There is \b[no faithful one-dimensional real representation of $C_4$], since a nonzero real scalar of finite order is $1$ or $-1$, and neither has order four. The two-dimensional representations above are real.
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