7.5 FK(metric designation 7.5x27mm) is a bottlenecked1 high power pistol cartridge which was meant to match the power & range of intermediate cartridges out of a pistol length barrel in an actual pistol. It partially succeeds at this as, when fired out of a 6 inch barrel, it manages only ~300J less than the russian 7.62x39mm and around ~200J more than 5.56x45mm (both) out of a 7.5in barrel.
Weapons chambered in the caliber include the FK BRNO Field Pistol and the FK PSD. Unfortunately, even though these weapons perform well at range and can pierce soft body armor, neither of these weapons are utilized by any military users(at least, not that I can see), likely due to cost and re-training/re-equipping concerns.
It is effective at piercing soft body armor, with both the flat tip and even the discarding hollow point being able to penetrate up to NIJ level IIIA2 body armor and there reportedly also being an armor piercing round for the caliber which can pierce level 4 body armor, although I highly doubt that they mean NIJ level IV since that armor is rated to stop .30-06 AP3. This same penetrative effect means it is unlikely to make a good home defense caliber with the ammo I've seen tested due to its tendency to overpenetrate, even piercing 2 long blocks of ballistic gel with a discarding hollow point round4. Although, Sierra bullets apparently has made a proper Lead core JHP meant to reduce penetration and get better energy transfer5.
See also:
.30 Carbine: Non-bottlenecked cartridge with similar bullet diameter and a bit more kinetic energy (although they might perform about the same out of a similar barrel length).
9x25mm Dillon: 10mm Auto necked down to 9mm. Round with close to as much kinetic energy but a wider(9mm) bullet.
7.62x25mm Tokarev: Round with similar bullet diameter but substantially less kinetic energy. Also known to be capable of piercing body armor.
1This means that the case(the bit that's made of brass or sometimes steel) gets thinner towards the bit with the bullet sticking out, this is done in order to fit a thinner bullet but have more propellant behind it.
2NIJ(National Institute of Justice) level IIIA body armor is rated to stop .357 Sig flat nosed FMJ and .44 Magnum hollow points. It is usually made from kevlar or other aramid fiber.
3.30-06 is a full power rifle round, so I don't believe that this pistol round, no matter how advanced, could've managed to out perform it against armor, especially out of the PSD's 5.3in barrel. Although, the M2 armor piercing round that the NIJ(National Institute of Justice) standard uses does date back to 1939 and, as one comment on Youtube said, "With enough engineering and tungsten, anything is possible". So it could be true, especially since the plate shown had only 1 penetration from 3 hits so it's possible the first 2 hits compromised the plate and then the third penetrated.
4What is a discarding hollow point round? It is a cartridge loaded with a bullet that has a hollow point that breaks off after impacting a target. Why would you ever want or need this? I have no clue! Please tell me if you know why
5Based on a test I've seen against a meat(to simulate ribs & muscle tissue) & oranges(to simulate fragile lung tissue) target, while its penetration was still somewhat overkill for defending against people who aren't wearing body armor, its terminal effect was devastating.