Think about what determines cast time for any fracture: the bone type, blood supply and stability.
The fracture: The base of the fifth metatarsal is cancellous (spongy) bone with a generous blood supply - the proximal tuberosity avulsion typically unites well, unlike the watershed Jones fracture a little more distally.
Healing biology: Cancellous metaphyseal bone with good vascularity heals in roughly 6-8 weeks. A below-knee (short-leg) cast or walking boot held for this window protects the fracture through the soft-callus and hard-callus phases until union is achieved.
Matching the options:
$\bullet$ 2-3 weeks - only the inflammatory/early phase is complete; union not yet reached.
$\bullet$ 3-5 weeks - still premature; high re-displacement risk.
$\bullet$ 16-20 weeks - excessive; reserved for slow-healing or non-union situations, and would stiffen the foot.
$\bullet$ 6-8 weeks - matches the natural union time of this fracture.
Therefore the correct immobilisation duration is 6-8 weeks (option A).