= Solution
Use $a$ for the acetyl methyl, $b$ for the O-bound <methylene group> pair, $c$ for the internal vinyl <proton>, and $d,e$ for the terminal vinyl <protons> respectively trans and cis to $c$. The assignments are
$$
\begin{array}{c|c|c|l}
\text{protons}&\delta_H/\mathrm{ppm}&\text{integral}&\text{resolved pattern}\\\hline
a&2.081&3&\text{singlet}\\
b&4.570&2&\text{doublet of triplets}\\
c&5.919&1&\text{doublet of doublets of triplets}\\
d&5.314&1&\text{doublet of apparent quartets}\\
e&5.235&1&\text{doublet of apparent quartets}
\end{array}
$$
The two terminal <protons> are <diastereotopic>: one is cis and the other trans to $c$, so they have different shifts and different <vicinal couplings>. Each terminal <proton> also couples through <geminal coupling> to the other and through <long-range scalar coupling> to the two $b$ <protons>. When those small couplings coincide, an underlying <doublet of doublets> of <NMR triplets> collapses to the observed <NMR doublet> of apparent <NMR quartets>. The O-bound <protons> are <enantiotopic> and have <chemical equivalence in NMR> in this achiral environment; they are split by $c$ and by the two terminal <protons>. The integrated terminal envelope contains two <protons>, not two equivalent <protons> at one shift.
\Image[/past-exam-of-the-chemistry-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1-allyl-couplings.png]
{title=Original stereochemically labelled proton structure for C with all resolved vicinal, geminal and long-range couplings}
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