How can a person who accused another of zina avert the hadd punishment from himself?

The hadd punishment for making an accusation of zina is
eighty lashes, because Allah, may He be exalted, says (interpretation of the
meaning):

“And those who accuse chaste women, and produce not four
witnesses, flog them with eighty stripes”

[an-Noor 24:4]. 

If one man accuses another of zina, and the accused person is
chaste (i.e., innocent), the hadd punishment must be carried out on the one
who made the accusation. 

The hadd punishment for making an accusation of zina may be
waived in a number of cases: 

1.

If he produces four witnesses who testify that the accused
man did commit zina, because Allah, may He be exalted, says (interpretation
of the meaning):

“And those who accuse chaste women, and produce not four
witnesses, flog them with eighty stripes”

[an-Noor 24:4]. 

2.

If the accused admits that he did commit zina. This is
according to scholarly consensus. See al-Mughni, 12/386. 

3.

If the accuser is the husband and has accused his wife of
zina, he may have avert the hadd punishment from himself by engaging in
li‘aan, because Allah, may He be exalted, says (interpretation of the
meaning):

“And for those who accuse their wives, but have no
witnesses except themselves, let the testimony of one of them be four
testimonies (i.e. testifies four times) by Allah that he is one of those who
speak the truth.

And the fifth (testimony) (should be) the invoking of the
Curse of Allah on him if he be of those who tell a lie (against her).”

[an-Noor 24:6-7]. 

When Hilaal ibn Umayyah accused his wife of zina, the Prophet
(blessings and peace of Allah be upon him) made them engage in li‘aan, and
did not carry out the hadd punishment on Hilaal for making an accusation of
zina. Narrated by Muslim, 1496 

4.

If the accused agrees to pardon the accuser and does not
demand that the hadd punishment be carried out on him. This has been
discussed previously in the answer to question no.
112116. 

Based on that, if a man accuses another man of zina and is
lying, he has no way of averting the hadd punishment from himself except by
apologising to the one whom he accused, and asking him to forgive him. If he
forgives him, then praise be to Allah, but if he wants to demand his right
of having the hadd punishment carried out, then he has the right to do
that. 

And Allah knows best.