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?)

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

In both cases, all the sentences convey the same general meaning that he admitted to cheating on the test. However, there are some subtle differences in the tenses used.

For sentences 1, 2, and 3:
- Sentence 1 (#1) means that he admitted to cheating on the test at the present moment when the statement is being made.
- Sentences 2 and 3 (#2 and #3) mean that he admitted to cheating on the test in the past, but they differ in terms of the tense used. Sentence 2 uses the simple past tense ("cheated"), while sentence 3 uses the past perfect tense ("had cheated").

For sentences 4, 5, and 6:
- Sentence 4 (#4) means that he currently admits to cheating on the test.
- Sentence 5 (#5) means that he admits to cheating on the test as a general habit or ongoing behavior.
- Sentence 6 (#6), like sentence 2, means that he admitted to cheating on the test in the past. However, sentence 6 uses the simple past tense ("cheated") while sentence 2 uses the past perfect tense ("had cheated").

So, to summarize:
- Sentence 1 means he admitted to cheating on the test currently.
- Sentences 2 and 3 mean he admitted to cheating on the test in the past, with a slight difference in the tense used.
- Sentence 4 means he currently admits to cheating.
- Sentence 5 means he generally admits to cheating on tests.
- Sentence 6 means he admitted to cheating on the test in the past.

In 1 and 2, "He admitted cheating on the test" and "He admitted that he cheated on the test" convey the same meaning. Both sentences indicate that the person acknowledged cheating on the test. The difference between them lies in their structure, with #2 using the phrase "that he cheated on the test" as a direct object of the verb "admitted."

In 3, "He admitted that he had cheated on the test" refers to a past action of cheating. The use of "had cheated" suggests that the cheating occurred before the admission took place. This structure denotes a sequence of events where the cheating happened first, and then the person admitted to it.

Regarding the last group of sentences, 4 and 5 carry contrasting meanings. "He admits cheating on the test" implies that the person confesses to currently cheating on the test. This sentence indicates an ongoing action. On the other hand, "He admits that he cheats on the test" implies a habitual or repeated action of cheating. It suggests that the person acknowledges that they frequently or consistently cheat on tests.

Finally, "He admits that he cheated on the test" in sentence 6 is structured similarly to sentence 2, indicating a past action of cheating. Just like sentence 3, this structure implies that the admission occurred after the cheating took place. Both 3 and 6 convey the same basic meaning with slightly different structures.

In summary:

- #1 and #2: Same meaning, indicating admission of cheating on the test.
- #3: Indicates past cheating before admission.
- #4: Indicates current cheating.
- #5: Indicates habitual or repeated cheating.
- #6: Similar meaning to #2 and #3, indicating past cheating before admission.