Question

1. He admitted cheating on the test.
2. He admitted that he cheated on the test.
3. He admitted that he had cheated on the test.

cf. He admitted having cheated on the test.

(Does #1 mean #2 or #3?)

4. He admits cheating on the test.
5. He admits that he cheats on the test.
6. He admits that he cheated on the test.
cf. He admits having cheated on the test.

(Does #4 mean #5 or #6?)

Answer 1

Only #5 is not quite correct. All others are fine, depending on when the cheating took place.