Upon entering the hospital's front door it is possible that Sinjin will immediately kill Kent. It is unclear what triggers this action.
Reload from a previous save and try again if you wish to save Kent (Power Armor known to cause issues, possibly caused by targeted HUD mod on helmet).
Additionally, Berry Mentats specifically will cause Kent to be killed. If you take Berry Mentats anytime before entering the room with Sinjin, Kent will be killed shortly afterwards.
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One should take extreme caution if they have the fourth rank of the Big Leagues perk, as attacking Sinjin with a melee weapon will most likely result in the player killing Kent by accident.
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It is possible to prevent Sinjin from killing Kent after intimidating his henchmen as the Shroud, by using Jet or entering VATS and shooting him in the head repeatedly, as this will stagger him and prevent him from shooting.
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Alternatively, taking Berry and Graqe Mentats (or simply one or the other) may cause the raiders surrounding Sinjin to fire at the player character, but keep Sinjin from doing anything at all.
This leaves the player free to, hopefully, give Sinjin a swift death while tanking rounds from his cohorts.
Following Sinjin's death, Kent will NOT be directly targeted, however, Kent can still be killed in the crossfire while you are fighting Sinjin's gang.