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Background
While talking to the deserter on the
Island (#1), he'll mention that he's used his rifle to kill people. When you ask him if he shot the hanged man, you'll trigger this task.
Walkthrough
To get the deserter to confess, you'll have to present at least three pieces of evidence linking him to the crime. This can include his gun (which you'll need to pick up after he drops it), the
Dried May Bells from the roof of the
Whirling-in-Rags, the footprints behind Klaasje's bedroom in the
Whirling-in-Rags, and the mattress in the sniper's nest on the
Island (#2).
To come up with enough evidence, you'll have to hit yourself in the head. This might cause your partner (Kim or Cuno) to pull you aside and give you a pep talk -- and also a hint about things you could still investigate.
Once you've presented enough evidence, the deserter will confess to the murder. This will complete the task and earn you 70 xp.
1 - Deserter
2 - Sniper's Nest