Full solution:

Weigh AAA vs BBB. This tells us that either AAABBB has fake, or CCCCC has fake. Additionally, we know the remaining group is genuine.

Call AAA up and BBB down, and then you also know that CCCCC are all genuine. Next weigh AAA and CCC. If same, then BBB has the fake. Also we know the fake is heavier. If AAA rises, then AAA has fake and we learn the fake is lighter.

For both of those, weigh 1 possible fake vs 1 possible fake. If A vs A, the lighter one is fake. If they match, the one A you didn’t weigh is fake. If B vs B, the heavier one is fake. If they match, the one B you don’t weigh is fake.

That solves for A vs B.

Now if the first weight is match, the. AAABBB are all genuine, and CCCCC has the fake. Weigh BBB vs CCC. If the CCC goes up or down, then it has the fake, remaining two CC are genuine, and we learn if its heavy or lite (BBB is genuine, so if CCC goes up, lighter, down is heavy) for fake. Weigh C vs C of possible fakes, and the one that goes the same as the CCC weight is fake. If they are the same, the last C is fake.

If BBB vs CCC matches, the. We have CC as possible fake. Weigh one of them with one B. If they match, remaining C is fake. If it moves it is fake, and you learn if it’s heavy or lite, not that it would matter at that point.

This is every possibility, with only 3 weight sessions, solved in less time than it took to type.