The following verse from Manu Smriti gives the impression that a woman can perform adultery when her husband is unwilling.
Manu Smriti 8.317. The killer of a learned Brahmana throws his guilt
on him who eats his food, an adulterous wife on her (negligent)
husband, a (sinning) pupil or sacrificer on (their negligent) teacher
(or priest), a thief on the king (who pardons him).
And, there are also the following verses from Atri Smriti, which too give the impression that a woman can commit adultery without accruing any sin.
A woman is not rendered guilty by her connection with her paramour,
»or a Brahmana by his performance of non-Vedic rites, nor a (river)
water, by urine and excreta ; nor fire, by burning impure articles.
(194). Women were first enjoyed by the Celestials ; then, by the moon,
the Gandharvas and the Fires. Afterwards came men to enjoy them. They
are never affected by any sin."
(195)
But, that impression is not correct. When women commit adultery but do not conceive, then they get rid of such a sin during their flow of menses. So, that sin is not a big issue. No special punishments are necessary as purification happens naturally.
But if conception happens, then penances are prescribed, because abortion can not be performed (as it is the most heinous crime) and if the progeny is born then that just adds to the confusion of castes. So, then it becomes a big issue and punishable offence.
See the following verse from Yajnvalkya:
II.—The purification from unchastity is at the season ; in case of
conception her abandonment is ordained. So also in case of causing
abortion or killing a husband or committing a heinous crime, and the
rest,
--72.
Yajnavalkya Smriti
So, the import is, if it's simply a case of adultery (minus conception, abortion) then that sin is automatically atoned for a woman. No other punishments are being required or prescribed.
In his commentary on the Yajnavalkya Smriti, Mitakshara, quotes from various scriptures to prove this point (that women committing adultery is never allowed by scriptures).
mitAksarA.
"The unholiness that arises from unrevealed mental chastity, I.e.,
from enjoying another man in imagination, is purified at the
*' season," i.e., menstruation. When conception is produced by (adulterous intercourse) with a friend, she should be abandoned.
CHAPTER III—MARRIAgE, v. LXXIL 139
As it has been said in Vasista-smriti (XXI. 12):-—" The wives of
Bralimanas, Ksatriyas, and Vaifiyas who commit adultery with a Sudra
may be purified by a penance in case no conception has taken place
(from their adulterous intercourse), not otherwise." So also in
"causing abortion," in '* killing the husband," and in " committing a
heinous crime," such as killing a Brahmana and in having adulterous
intercourse with a pupil, &c., (this being implied by the use of the
word *' and the rest " in the text) she should be abandoned. As it has
been ordained by VYASA (Vasistha XXI. 10.) "But these four wives must
be abandoned, viz.^ one who yields herself to her husband's pupil, or
to his Guru, and especially one who attempts the life of her lord, and
who commits adultery with a man of a degraded caste (Junghita)." The
Junghitas (outcastes) are the descendants of Pratilomas, such as
leather-tanners &c. The abandonment should be by not having any carnal
connection with her, and by not allowing her to join in any religious
ceremonies and does not mean that she should be driven out of the
house, because of the rule " she should be kept confined to one
apartment " (Manu XL 176 or 177).
" If (a wife) has been mentally unfaithful to her husband, she shall
live on barley or rice boiled in milk during three days, and sleep on
the bare ground. After the three days (have expired), the (husband)
shall offer eight hundred burnt oblations, (reciting) the Savitr (and
the Mantra called) Siras, while she is immersed in water. It is
declared in the Veda that she becomes pure (thereby),
*' If (a wife) has held an (improper) conversation (with another man), she must perform the same penance during a month. After (the
expiration of) the month, (the husband) shall offer four times eight
hundred burnt-oblations, (reciting) the Savitri (and the Mantra
called) Siras, while she is immersed in water. It is declared in the
Veda that she becomes pure (thereby)." The meaning of the word "
Aprajata " in the above verse of Vasistha is, " when no conception has
taken place" (and it does not mean, " when no child is born,"
balambhatta' gloss.
In the case of revealed mental unfaithfulness, the rule of three days'
fast &c., as laid down in Vasistha (as given in the above note) should
be observed. In case of unrevealed mental unfaithfulness, there is no
open penance, and so Vijnanesvara says " unrevealed," t.e., not known
to any second person (not even to her husband, that she has been
mentally unchaste). This mental unchastity consists in thinking to
have intercourse with another person. Therefore Vijnanesvara uses the
milder term, " unholiess," and not the harsher term, " sin." The wives
may be abandoned in the four cases mentioned by VYaSA, also in some
other cases mentioned by Vasistha.
So also Devala and Narada (XII. 90-94) ;— *• When husband and wife
leave one another, from mutual dislike, it is a sin, except when a
woman, who is kept under supervision, commits adultery. *' When a
married woman commits adultery, her hair shall be shaved, she shall
have to lie on a low couch, receive bad food and bad clothing, and the
removal of the sweepings shall be assigned to her as her occupation.
One who wastes the entire property of her husband under the pretence
that it is (her own) Stri-dhana, or who procures abortion, or who
makes an attempt on her husband's life, he shall banish from the town.
One who always shows malice to him, or who makes unkind speeches, or
eats before her husband, he shall quickly expel her from his house. *'
Let not a husband show love to a barren womatt, or to one who gives
birth to female children only, or whose conduct is blamable, or who
constantly contradicts him ; if he does (have conjugal intercourse
with her), he becomes liable to censure (himself)."
So also Yama :— '* A wife who does not approach her husband in
season, and thus kills the embryo, should be exiled, after proclaiming
her through the village. She who owing to hating her husband does not
approach him in season must be proclaimed to the kinsmen, and exiled
since she is a killer of the foetus."
So also Baudhayana :—" Wise men quickly abandon that wife who is
illtongued, undutiful, barren or unchaste, and hating her husband. A
wife, who does not give birth to a child, should be renounced in her
tenth year of marriage, and in her twelfth year, if she gives birth to
female children only, and in her fifteenth year, if she gives birth to
dead children only, but a wife must be renounced at once» if she is
unpleasant-speeched."
So also Harita :— " Let him abandon the wife who kills the embryo,
or commits adultery with a person of low caste, or with a pupil, or
with one related to her as a son, or who is addicted to drink and
other vices, or who wastes wealth and property."
So also Yama :—" The abandonment of that wife is enjoined, who is
independent in her movement. A woman should never be killed, nor
deformed by mutiliation. Manu-Vaivasvata has enjoined the renouncement
of the wife, who is independent and adulterous. He has also ordained
that a woman should never be killed, nor deformed."